To describe: The meaning for f to be differentiable at a.
If a is appoint in the domain of a function f, then f is said to be differentiable at a if the derivative exists. This means the graph of f has non-vertical tangent line at the point . If f is differentiable at a point a then f must be continuous at a.
To describe: The relation between differentiability and continuity.
All differentiable functions are continuous, but not all the continuous functions are differentiable.
Let a function is differentiable at a point in its domain only if is continuous at .
Function is differentiable at a point is
Since, x approaches implies , then
Take limit as x approaches to of both sides
Since there exist and is contained in the domain of f, is continuous at .
To sketch: A function that is continuous but not differentiable at .
Let the function and f is continuous but it is not differentiable at 2.
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