Question

Does every point on a differentiable function have a tangent line?

Answer 1

Yes. Assuming that I understand your question correctly.

Explanation:

If function `f` is differentiable at `a`, then there is a line tangent to the graph of `f` at the point `(a, f(a))`.

If `f` is differentiable at `a`, then `f(a)` exists, and, by definition, the slope of the tangent line at `(a, f(a))` is `f'(a)`