Mean value Theorem :

If  is continuous on  and differentiable on open interval , then there exists a number  in  such that .

The function  is continuous at the point  in its domain if :

1.  exists,

2. .

Now observe the graph :

Here  and .

So .

The function is continuous on .

Check for differentiability.

The left hand limit and right hand limits are different since the there are two different function on the left hand side and right hand side at .

So the function is not differentiable at .

The function is not differentiable on the closed interval .

Therefore the mean value theorem does not apply to the function.

The mean value theorem does not apply to the function.