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 f ' (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 ∞:
Approaching from the right, the slopes of the secant lines approach -∞:
Since the one-sided limits disagree,
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.
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.