What are differentiable points for a function?

1 Answer
George C.
Dec 3, 2015

A differentiable point of a function #f(x)# is a value #a# such that the two-sided limit exists and is finite:

#lim_(h->0) (f(a+h)-f(a))/h#

Explanation:

For example, let us verify that #f(x) = x^2 + x# is differentiable at #x=2#:

Let #a = 2#

Then:

#lim_(h->0) (f(a+h)-f(a))/h#

#=lim_(h->0) (f(2+h)-f(2))/h#

#=lim_(h->0) (((2+h)^2+(2+h))-(2^2+2))/h#

#=lim_(h->0) ((4+4h+h^2+2+h)-(4+2))/h#

#=lim_(h->0) (5h+h^2)/h#

#=lim_(h->0) (5+h)#

#=5#

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