Question

When does a function is called continuously differentiable?

Answer 1

A differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps.

Answer 2

A function is called continousely differentiable if the derivative of the function exist at it is continous at each point of the domain .

That means if find the limit for the derivative of the function from left side and the right side, it should be same to the derivative at that point .

It is not necessary that a continous function will be differentiable also OR if the function is continous at some point then it's derivative os also continous at that point .

For example the function f(x)= 4 is continous function but not differentiable

Another example f(x)= x^2 sin(1/x) if x is not 0

f(x)= 0 if x= 0

here derivative of f(x) exist at x=0 but if we find f'(x) at the limit x---->0 it does not exist

so the derivative is not continous so this function is not differentiable continously .