Continuity

A function $f(x)$ is continuous at $x = a$ if the following three conditions are satisfied:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

Formally,

Note: being continuous at a specific point and being a continuous function are different things. A function is continuous if it is continuous at every point in its domain.

Differentiability

A function $f(x)$ is differentiable at $x = a$ if the following limit exists:

Here, $f'(a)$ is called the derivative of $f(x)$ at $x = a$.

Note: again, ‘differentiable at a point $a$’ and ‘differentiable function’ are different things. A function is differentiable if it is differentiable at every point in its domain.

The relationship between continuity and differentiability

If a function is differentiable at a point, then it is continuous at that point. However, the converse is not necessarily true.