In previous section we have discussed that if a function is differentiable at a point, then it should be continous at that point and a discontinuous function cannot be differentiable. This fact is proved in following theorem
Theorem: If a function is differentiable at a point, it is necessarily continuous at that point.But the converse is not necessarily continuous true.
f(x) is differentiable at x=c f(x) is continuous at x=c
In order to prove that f(x) is continuous at x=c, it is sufficient to show that
Hence f(x) is continuous at x=c
Converse: The converse of above theorem is not necessarily true i.e., a function may be continuous may be continuous at a point but may not be differentiable at that point.
For example: The function f(x) =|x| is continuous at x=0 but it is not differentiable at x=0, as shown in the figure.
Which shows we have sharp edge at x=0 hence, not differentiable but continuous at x=0