Linearization and approximation

Question

If f is continuous at x = c, then f is differentiable at x = c.

Options:

1. true
2. false

Answer 1

False

Here is a counter example

y = f(x) = |x| = sqrt(x^2) = (x^2)^(1/2)

y' = (1/2)(x^2)^(-1/2) ( 2x) = -x/x^2 = -1/x

the derivative does not exist at x=0

therefore the function is not differentiable at x=0

Continuity does NOT guarantee differentiability..

However, differentiability does guarantee continuity