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At a Glance - Differentiability and Continuity

We've had all sorts of practice with continuous functions and derivatives. Now it's time to see if these two ideas are related, if at all.

We say a function is differentiable at a if f ' (a) exists. A function is differentiable on an interval if f ' (a) exists for every value of a in the interval. We say a function is differentiable (without specifying an interval) if ' (a) exists for every value of a.

A differentiable function must be continuous.

In other words, a discontinuous function can't be differentiable. Think of all the ways a function f can be discontinuous.

  •  f(a) could be undefined for some a.

We already said that if a function isn't defined at a, we can't find its derivative at a, since the calculation of the derivative uses f(a).

  • could fail to exist. In this function, the one-sided limits disagree. As a result, the secant lines approach a from the left aren't approaching the same thing if we approach a from the right.

This means we can't draw a tangent line to f at a, so f ' (a) doesn't exist, and f isn't differentiable.

This function is another example:

  •  could disagree with f(a). In this case, the secant lines between a and a + h have negative slope when h is positive, and positive slope when h is negative:

Approaching from the left, the slopes of the secant lines approach ∞:

This means

Approaching from the right, the slopes of the secant lines approach -∞:

Since the one-sided limits disagree,

 doesn't exist.

A continuous function doesn't need to be differentiable.

There are plenty of continuous functions that aren't differentiable. Any function with a "corner" or a "point" is not differentiable.

Sample Problem

f(x) = |x| is not differentiable because it has a "corner" at 0.

Differentiable functions are "smooth," without sharp or pointy bits. Dragon's teeth, either the teeth of a fire-breathing dragon or the fortifications used in WWII, would not be differentiable.

Exercise 1

Determine whether each of the following functions is (a) continuous, and (b) differentiable.

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