Continuity

The notion of Continuity is one of the most important in real analysis, partly because continous functions most closely resemble the behaviour of observables in nature.

Although continuity and continous functions can be defined on more general sets, we will first restrict ourselves to $\mathbb{R}$

Definition

Let $A\subset\mathbb{R}$

Let $f:A\rightarrow\mathbb{R}$

Let $c\in A$

We say that $f$ is continous at point $c$ if $\forall\varepsilon>0\;\exists\delta>0$ such that for all $x\in A$ , $|x-c|<\delta\Rightarrow |f(c)-f(x)|<\varepsilon.$

If $f$ is continous at $c$ for all $c\in A$ , we say that $f$ is continous over $A$ .

Definition for metric spaces

We can easily extend this definition to metric spaces. Let $X$ and $Y$ be metric spaces. Given a function $f:X\to Y$ , and a point $c\in X$ , we say that $f$ is continuous a $c$ if, for all $\varepsilon >0$ there is a $\delta>0$ such that for all $x\in X$ , $d_X(c,x)<\delta\Rightarrow d_Y(f(c),f(x))<\varepsilon.$

If $f$ is continous at $c$ for all $c\in X$ , we say that $f$ is continous over $X$

Definition for Topological spaces

Perhaps the most general definition of continuity is in the context of topological spaces. If $X$ and $Y$ are topological spaces, then a function $f:X\to Y$ is called continuous if for any open set $\mathcal{U}$ in $Y$ , it's preimage (i.e. the set $f^{-1}(\mathcal{U}) = \{x\in X|f(x)\in\mathcal{U}\}$ ) is an open set in $X$ . Note that the image of an open set in $X$ does not have to be open.

It can be shown that if $X$ and $Y$ are metric spaces under the metric space topology, that this definition of continuity coincides with the previous one.

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Continuity

Definition

Definition for metric spaces

Definition for Topological spaces