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 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 $c$ iff $\forall\epsilon>0\;\exists\delta>0$ such that $x\in V_{\delta}(c)\implies f(x)\in V_{\epsilon}(f(c))$

If $f$ is continous at $c$ $\forall c\in A$ , we say that $f$ is Continous over $A$

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Continuity

Definition