# Differentiability and Continuity

If f(x) is differentiable at every point of its domain, then it must be continuous in that domain.

Notes:
• The converse of it is false; that is, there are functions that continuous but not differentiable. For instance, the function f(x) = | x | is continuous at 0 because
• The converse of the above result is not true, i.e., if f(x) is continuous at x = a then it may or may not be differentiable at x = a.
• If f(x) is differentiable then its graph must be smooth, i.e., there should be no break or corner.

Thus for a function f(x):
• Differentiable ⇒ Continuous
• Continuous ⇒ May or may not be differentiable
• Not continuous ⇒ Not differentiable

# How can a function fail to be differentiable?

The function f(x) is said to be non differentiable at x = a, if
1. both Rf′(a) and Lf′(a) exist but are not equal
2. either or both Rf′(a) and Lf′(a) are not finite
3. either or both Rf′(a) and Lf′(a) do not exist
The function y = |x| is not differentiable at 0 as its graph changes direction abruptly when x = 0. In general, if the graph of a function has a “corner” or “kink” in it, then the graph of f has no tangent at this point and f is not differentiable there. [In trying to compute f′(a), we find that the left and right limits are different.]

If f is not continuous at a then f is not differentiable at a. So, at any discontinuity (for instance, a jump discontinuity) f fails to be differentiable.

A third possibility is that the curve has a vertical tangent line when x = a, that is, f is continuous at a and .

This means that the tangent lines become steeper and steeper as x a. Following figure illustrates the three possibilities that we have discussed.