How much you know calculus
Calculus
Level
4
I define a function h(x) which is continuous over it's domain.
Given that Derivative of h(x) is discontinuous at point P. What you could conclude?
Function is surely differentiable at P
Function may or may not differentiable at P
Function is surely non differentiable at P
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1 solution
First of all, differentiability is nothing to do with the derivative of a function.
Differentialbility is totally a different criteria. One has to use first principle in order to check differentiability. If the derivative of function is discontinuous at a given point, it doesn't implies that function is non differentiable at that point unless first principle says so.
I am using mobile hence it is been a problem in typing hence I can't give a example right now. But, I can give you the function
(x^2)*(sin(1/x)) for x not equal to zero and 0 for x=0
If you will check, you will find that derivative is discontinuous at x=0 , but if you check differentiability using first principle , you will find that function is differntiable at x=0.
I will update this solution and upload a better solution when I will get my laptop (Probably on Saturday). Hoping that someone will post a better solution till then.