Definition:Continuous Real Function

Definition
Let $f: \R \to \R$ be a real function.

Then $f$ is continuous on $\R$ iff $f$ is continuous at every point of $\R$.

Continuity on an Interval
Where $A$ is a real interval, it is considered as a specific example of continuity on a set.

It is worth addressing each type of interval in turn.

As a Metric Space
Note that the definition for continuity at a point as given here is the same as that for a metric space, where the usual metric is taken on the real number line.

Informal Definition
The concept of continuity makes precise the intuitive notion that a function has no "jumps" or "holes" at a given point.

Loosely speaking, a real function is continuous at a point if the graph of the function does not have a "break" at the point.