Derivative And Continuity

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\todo }[0]{} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\sagemath }[0]{\textsf {SageMath}} \DeclareMathOperator {\dx }{dx} \DeclareMathOperator {\dt }{dt} \DeclareMathOperator {\dy }{dy} \DeclareMathOperator {\dz }{dz} \DeclareMathOperator {\dr }{dr} \DeclareMathOperator {\dw }{dw} \DeclareMathOperator {\du }{du} \DeclareMathOperator {\dv }{dv} \DeclareMathOperator {\ds }{ds} \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\point }[1]{\left (#1\right )} \newcommand {\pt }[1]{\mathbf {#1}} \newcommand {\Lim }[2]{\lim _{\point {#1} \to \point {#2}}} \newcommand {\bar }[0]{\overline } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

We discuss how continuity and differentiability are interrelated.

Video Lecture

Text with Additional Details

One important, but easy to overlook question of derivatives, is when the derivative actually exists. It may seem strange, since derivatives are often referenced as instantaneous rate of change, or the rate of change at a point, so it might seem intuitive that as long as the function exists at a point, the derivative should too. Unfortunately this is not the case, and like so many other aspects of derivatives, it comes down to the fact that the derivative is actually a limit in disguise.

We start with a bit of terminology:

Differentiable A function is said to be differentiable at $x=a$ , if the derivative exists at $x=a$ . A function is simply said to be differentiable if the derivative of $f(x)$ exists for every $x$ value in the domain of $f$ .

We want to know when a derivative is defined, or in other words, when it exists. Since the derivative is actually a limit of a difference quotient, what we really mean when we ask if a derivative exists, is whether or not the corresponding limit exists. This is a little bit tricky though, because we aren’t asking about the limit of the function, rather we are asking about the limit of the difference quotient. So, a good way to think about whether or not a derivative exists at a given point, is to determine what is happening with the rate of change. There are two things to consider here, and we will start with the most obvious.

Consider the following graph:

Now, on the one hand, this graph is relatively simple, it is a shift of the classic $\frac {1}{x}$ graph. If we wanted to look at a tangent line at, say, $x=0$ , we could figure it out (with a little bit of effort) and indeed we can see that it works here:

But, what if we wanted a tangent line for $x=1$ ? After a moments investigation we can see we have a problem, the function doesn’t even exist at $x=1$ ! And, somehow even worse than that, if we go a little bit to the right and a little bit to the left, the “nearby” points end up about as far away from each other as possible; one being arbitrarily large and positive and the other being arbitrarily large and negative!

This leads us to our first observation. In order for a derivative to exist, we need the function to exist. After all, it’s difficult to talk about being tangent to a graph at a specific point (meaning, touching the graph at a single point), if the graph doesn’t even exist at that point!

Ok... so the function needs to exist at the point, but that was probably not a huge revelation; that makes sense after all. But we did say there were two things to consider. So what’s the other one?

For our second observation, let’s consider trying to make a tangent line at $x=0$ on the following graph:

At first it may seem like we should be ok trying to make the tangent line; after all, we know at least where it should be tangent to. But after a bit of experimentation, we can quickly see that we have a problem choosing which tangent line is the correct tangent line. Indeed, consider the following examples:

We can see that the line only touches the graph in one spot, it is indeed (graphically) tangent. As is this one:

And this one:

And this one...

Take a moment and see if you can figure out why we are able to have so many different tangent lines for the same spot.

As you may have determined, the problem here is the gap in the graph. We need there to be no gap in order to force the tangent line to be unique (the algebra involved here is a bit tedious and involved, but not too complicated; it is worth trying it yourself!) A more eloquent way of describing this is that we need the limit to exist; this is our second observation.

But wait! If we need the function to exist, and we need the limit to exist... and we clearly need them to be the same value in order to not have a problem with our tangent line... this is exactly the same as saying we need the function to be continuous!

So, it appears that, in order for the derivative to exist, we need the function to be continuous. In other words it is necessary that the function is continuous at the desired point. But the obvious follow up question is... is it sufficient for the function to be continuous, or do we need anything else?

As a side note, this is vital terminology in mathematics; sufficient versus necessary conditions!

Necessary conditions are conditions that must be met in order for something to be true, but a list of some necessary conditions may not include everything that is necessary. For example, having access to electronics is necessary to access the internet, but you also need some kind of internet service provider and a signal (either via wifi, mobile tower, satellite, or physical cable) as well. So the electronics are a necessary condition, but it isn’t enough on it’s own to have access to the internet.

In contrast, sufficient conditions are a (list) of conditions that suffice to make something true, but that list could include things that aren’t necessary. For example, winning a billion dollar lottery is a sufficient amount of money to buy a new car, but it’s definitely not necessary to win a billion dollar lottery to buy a new car.

A common pursuit in mathematics then, is to find a list of conditions that is both necessary and sufficient. This would mean the list includes everything that is needed (because it is sufficient) and nothing that is unneeded (because it is necessary).

Ok, back to our continuous versus differentiable discussion!

Unfortunately it isn’t quite enough to have the function be continuous. Consider, for example, the following graph.

What happens when we try to take tangent lines at the $x=-1$ point? We can see several tangent lines would work!

Like this one;

Or this one

Or this one...

Again, with a little thought we can see that the issue is the fact that we have this weird corner happening at the point $(-1,-3)$ . This is one of the two common ways that a continuous function can still have a non-differentiable point - that is, a point where the derivative fails to exists. This is, oh-so-cleverly, called a “corner” and is characterized by the two straight lines ending in a sharp point where the graph suddenly changes directions, notably without any kind of curve or smooth transition.

The other type of point where a function fails to be differentiable is called a cusp, and is similar to a corner, but the lines forming the sharp turn curve (in the wrong direction) as they meet. Consider the point $(2,1)$ of the following graph as an example.

We can see how the same method of putting many different tangent lines through the same point would work here.

What corners and cusps both have in common, is that the limit of the difference quotient fails to exist because the slopes of the function are very different as they get closer and closer to the corner or cusp.

Indeed, an actual term used by mathematicians (which has a rigorous meaning, but we’ll skip that for the sake of everyone’s sanity) when discussing graphs that are nicely differentiable, is “smooth” because it fairly accurately describes a function that is differentiable (there’s a little bit more happening here in terms of higher derivatives, but again that’s not important for the sake of our discussion). Specifically, a function that is truly “smooth” will almost certainly be differentiable, but a function that is noticeably jagged or not “smooth” a point (like our corner or cusp example) will almost certainly not be differentiable. You always want to check analytically when possible to verify, but this gives a nice geometric litmus test for picking out points that might be problematic for a derivative when you have access to the graph.

So now we have seen how, with careful consideration, we can determine that a function must be continuous at a point in order for there to be any hope of a derivative existing at that point (in other words, continuity is a necessary condition). However, we have also seen that it is possible to be continuous at a point, and still not be differentiable at that point (in other words, continuity is not a sufficient condition); the two most common cases of this occurring are corners and cusps. So we can say that continuity at a point is necessary, but not sufficient, for a function to be differentiable at that point.