Question

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

Answer 1

No. A function can be continuous without being differentiable.

Explanation:

For example, the function `f(x) = abs(x)` is continuous on the whole of `RR`, but has no tangent at `x = 0`.

There are well known examples of functions that are continuous everywhere and differentiable nowhere. For example, the Weierstrass Function and Bolzano's Function.