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Derivatives

Derivatives

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.

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