## Theorem 6.5.5: Differentiable and Continuity

If

*f*is differentiable at a point*c*, then*f*is continuous at that point*c*. The converse is not true.### Proof:

Note that

- f(x) - f(c) = ( x - c )

As x approaches c, the limit of the quotient exists by assumption and is equal to f'(c), and the limit of the right-hand factor exists also and is zero. Therefore:

- f(x) - f(c) = 0

which is another way of stating that f is continuous at x = c.