Continuity and Differentiability

Proposition (Differentiability Implies Continuity) If a function is differentiable at then it is also continuous at .

    Proof. Assume that is a differentiable function, then by definition,



exists, and therefore we can use the product rule for limits with,











Whence, and so is continuous at

Example (Differentiability Implies Continuity) The following graphs illustrate the relationship between continuity and differentiability.



For example, the graph of the left is continuous at every real number but this graph is not differentiable at The graph of the right is not continuous at so it certainty is not differentiable there.

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Continuity And Differentiability
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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