In other words, a function f(x) is differentiable if and only if its graph is a smooth continuous curve with no sharp corners (a sharp corner would be a place where there would be two possible tangent vectors).

How do you know if a function is differentiable?

A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean? It means that a function is differentiable everywhere its derivative is defined. So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.

Does differentiability imply existence?

If a function is differentiable then it's also continuous. This property is very useful when working with functions, because if we know that a function is differentiable, we immediately know that it's also continuous.