Question

If there was a hole in the line at (2,3) and there is another point at (2,1), then would the graph be differentiable at that point and why?

Answer 1

If I understand the description correctly, the answer is no and the reason is below.

Explanation:

The short answer is that the function you have described is not continuous at `2`. It is a theorem that if `f` is differentiable at `c`, then `f` is continuous at `c`. Therefore non-continuous implies non-differentiable.

Longer answer

"A hole in the line at `(2,3)`" indicates to me that `lim_(xrarr2)f(x) = 3`.

The point at `(2,1)` implies that `f(2)=1`

Now
`f'(2) = lim_(xrarr2)(f(x)-f(2))/(x-2)`

`= lim_(xrarr2)(f(x)-1)/(x-2)`

This limit has the form `(3-1)/(2-2) = 2/0` which entails that the limit does not exist.

Since the derivative is the limit, the derivative also does not exist.