Continuity and Differentiability

Continuity and Differentiability. 12 class math full chapter 
What is the difference between continuity and differentiability?

Differentiability means that the function has a derivative at a point. Continuitymeans that the limit from both sides of a value is equal to the function's value at that point.

Does continuity mean differentiability?
If f is differentiable at a point x0, then f must also be continuous at x0. In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable.

Does continuity imply differentiability?
Differentiability Implies Continuity If is a differentiable function at , then is continuous at . ... If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

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