WEBVTT

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- [Instructor] What we're
going to do in this video

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is talk about the various
types of discontinuities

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that you've probably seen
when you took algebra,

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or precalculus, but then
relate it to our understanding

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of both two-sided limits
and one-sided limits.

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So let's first review the
classification of discontinuities.

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So here on the left,
you see that this curve

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looks just like y equals x squared,

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until we get to x equals three.

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And instead of it being three squared,

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at this point you have this opening,

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and instead the function at
three is defined at four.

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But then it keeps going
and it looks just like

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y equals x squared.

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This is known as a point,

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or a removable, discontinuity.

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And it's called that for obvious reasons.

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You're discontinuous at that point.

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You might imagine defining
or redefining the function

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at that point so it is continuous,

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so that this discontinuity is removable.

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But then how does this
relate to our definition

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of continuity?

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Well, let's remind ourselves
our definition of continuity.

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We say f is continuous,

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continuous,

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if and only if,

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or let me write f continuous

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at x equals c, if and only if

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the limit as x approaches c

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of f of x is equal to the
actual value of the function

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when x is equal to c.

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So why does this one fail?

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Well, the two-sided limit actually exists.

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You could find, if we say
c in this case is three,

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the limit

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as x approaches three

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of f of x,

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it looks like, and if you
graphically inspect this,

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and I actually know this is the
graph of y equals x squared,

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except at that discontinuity
right over there,

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this is equal to nine.

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But the issue is, the way
this graph has been depicted,

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this is not the same thing
as the value of the function.

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This function

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f of three, the way it's been graphed,

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f of three is equal to four.

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So this is a situation where
this two-sided limit exists,

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but it's not equal to the
value of that function.

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You might see other
circumstances where the function

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isn't even defined there,

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so that isn't even there.

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And so, once again, the limit might exist,

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but the function might
not be defined there.

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So, in either case, you aren't
going to meet this criteria

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for continuity.

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And so that's how a point
or removable discontinuity,

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why it is discontinuous

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with regards to our limit
definition of continuity.

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So now let's look at this second example.

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If we looked at our
intuitive continuity test,

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if we would just try to trace this thing,

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we see that once we get to x equals two,

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I have to pick up my
pencil to keep tracing it.

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And so that's a pretty good
sign that we are discontinuous.

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We see that over here as well.

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If I'm tracing this function,
I gotta pick up my pencil to,

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I can't go to that point.

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I have to jump down here,

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and then keep going right over there.

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So in either case I have
to pick up my pencil.

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And so, intuitively, it is discontinuous.

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But this particular type of discontinuity,

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where I am making a jump from one point,

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and then I'm making a jump
down here to continue,

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it is intuitively called a jump

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discontinuity,

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discontinuity.

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And this is, of course, a
point removable discontinuity.

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And so how does this relate to limits?

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Well, here, the left and
right-handed limits exist,

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but they're not the same thing,

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so you don't have a two-sided limit.

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So, for example, for
this one in particular,

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for all the x-values up to
and including x equals two,

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this is the graph of y equals x squared.

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And then for x greater than two,

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it's the graph of square root of x.

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So in this scenario,

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if you were to take the limit

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of f of x

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as x approaches

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two

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from the left,

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from the left,

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this is going to be equal to four,

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you're approaching this value.

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And that actually is the
value of the function.

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But if you were to take the
limit as x approaches two

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from the right of f of x,

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what is that going to be equal to?

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Well, approaching from the right,

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this is actually the square root of x,

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so it's approaching
the square root of two.

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You wouldn't know it's
the square root of two

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just by looking at this.

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I know that, just because when I,

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when I went on to Desmos
and defined the function,

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that's the function that I used.

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But it's clear even visually

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that you're approaching
two different values

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when you approach from the left

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than when you approach from the right.

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So even though the one-sided limits exist,

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they're not approaching the same thing,

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so the two-sided limit doesn't exist.

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And if the two-sided limit doesn't exist,

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it for sure cannot be equal to the value

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of the function there, even
if the function is defined.

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So that's why the jump
discontinuity is failing this test.

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Now, once again, it's intuitive.

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You're seeing that, hey, I gotta jump,

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I gotta pick up my pencil.

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These two things are not
connected to each other.

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Finally, what you see here is,

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when you learned precalculus,

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often known as an
asymptotic discontinuity,

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asymptotic,

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asymptotic

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discontinuity,

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discontinuity.

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And, intuitively, you
have an asymptote here.

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It's a vertical asymptote at x equals two.

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If I were to try to trace the graph

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from the left,

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I would just keep on going.

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In fact, I would be doing
it forever, 'cause it's,

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it would be infinitely,

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it would be unbounded as
I get closer and closer

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to x equals two from the left.

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And if try to get to x
equals two from the right,

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once again I get unbounded up.

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But even if I could,

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and when I say it's unbounded,
it goes to infinity,

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so it's actually impossible

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in a mortal's lifespan to
try to trace the whole thing.

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But you get the sense that,
hey, there's no way that I could

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draw from here to here
without picking up my pencil.

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And if you wanna relate it
to our notion of limits,

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it's that

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both the left and right-handed
limits are unbounded,

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so they officially don't exist.

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So if they don't exist, then
we can't meet these conditions.

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So if I were to say,

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the limit

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as x approaches two from the
left-hand side of f of x,

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we can see that it goes unbounded
in the negative direction.

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You might sometimes see someone
write something like this,

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negative infinity.

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But that's a little
handwavy with the math.

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The more correct way to say
it is it's just unbounded,

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unbounded.

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And, likewise, if we
thought about the limit

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as x approaches two

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from the right

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of f of x,

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it is now unbounded
towards positive infinity.

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So this, once again,

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this is also,

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this is also unbounded.

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And

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because it's unbounded and
this limit does not exist,

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it can't meet these conditions.

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And so we are going to be discontinuous.

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So this is a point or
removable discontinuity,

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jump discontinuity, I'm jumping,

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and then we have these
asymptotes, a vertical asymptote.

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This is an asymptotic discontinuity.