Definition of continuity

The intuitive idea of a “continuous” function is that of a graph that can be drawn without ever lifting the pen from the paper. To make this idea precise, three conditions are needed, all linked to the behaviour of the function near a point.

Definition — Function continuous at a point

A function $f$ is continuous at a point $x_0$ of its domain if and only if simultaneously three conditions hold:

1. $f(x_0)$ is defined (the point belongs to the domain);
2. $\lim_{x\to x_0}f(x)$ exists and is finite;
3. $\lim_{x\to x_0}f(x) = f(x_0)$ (the limit coincides with the value).

If even a single one of the three conditions is not satisfied, $f$ has a point of discontinuity at $x_0$.

The three conditions must be read in order: first the point must exist in the domain, then the limit must exist and be finite, finally the two values — that of the limit and that of the function — must coincide. It is enough for a single one to fail for continuity to be lost.

Links

Topics: Continuita
Concepts: Continuita · Discontinuita · Limite