# What is a continuous function?

Jul 9, 2015

There are several definitions of continuous function, so I give you several...

#### Explanation:

Very roughly speaking, a continuous function is one whose graph can be drawn without lifting your pen from the paper. It has no discontinuities (jumps).

Much more formally:

If $A \subseteq \mathbb{R}$ then $f \left(x\right) : A \to \mathbb{R}$ is continuous iff

$\forall x \in A , \delta \in \mathbb{R} , \delta > 0 , \exists \epsilon \in \mathbb{R} , \epsilon > 0 :$

$\forall {x}_{1} \in \left(x - \epsilon , x + \epsilon\right) \cap A , f \left({x}_{1}\right) \in \left(f \left(x\right) - \delta , f \left(x\right) + \delta\right)$

That's rather a mouthful, but basically means that $f \left(x\right)$ does not suddenly jump in value.

Here's another definition:

If $A$ and $B$ are any sets with a definition of open subsets, then $f : A \to B$ is continuous iff the pre-image of any open subset of $B$ is an open subset of $A$.

That is if ${B}_{1} \subseteq B$ is an open subset of $B$ and ${A}_{1} = \left\{a \in A : f \left(a\right) \in {B}_{1}\right\}$, then ${A}_{1}$ is an open subset of $A$.