Theorem: 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.