6.5. Differentiable Functions

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.


Note that

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:

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

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