Question

When is a function said to be differentiable?

Answer 1

when it is continuous at all points in defined interval
I. e.,
LHL=RHL=AT POINT
F(A+)=F(A)=F(A-)
AND,, graph of function must not contain any sharp edges or peaks..

Answer 2

For a function to be differentiable its left hand limit and right hand limit must same.

Right hand limit=RHL=f(a+)
Left hand limit =LHL=f(a-)

[Here we are checking limit at x=a]

=>For differentiability, RHL=LHL

If graph is not continuous, it is not differentiable at those points. This is evident by the fact that LHL must be equal to RHL for differentiability.

=>The graph must not have a sharp peak.

An example is given in the attachment for your understanding.
At there, graph is not differentiable at x=a (for discontinuity)

x=b, x=c, there are sharp peaks. So, it is also not differentiable at those points.

The function is differentiable at all other points
(like x=d, x=e)