Can a graph be differentiable at a specific point but not continuous at the same point?

Not according to the usual definitions of "differentiable" and "continuous".

Suppose that the function f is differentiable at the point x = a.

Then f(a) is defined and

limit (h -> 0) [f(a+h) - f(a)]/h exists (has a finite value).

If this limit exists, then it follows that

limit (h -> 0) [f(a+h) - f(a)] exists and equals 0.

Hence limit (h -> 0) f(a+h) exists and equals f(a).

Therefore f is continuous at x = a.