Category:Continuous Real Functions

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This category contains results about Continuous Real Functions.


Continuity at a Point

Let $A \subseteq \R$ be any subset of the real numbers.

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

Let $x \in A$ be a point of $A$.


Definition by Epsilon-Delta

Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and:

$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$


Definition by Neighborhood

Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and:

$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$
for every $\epsilon$-neighborhood $N_\epsilon$ of $\map f x$ in $\R$, there exists a $\delta$-neighborhood $N_\delta$ of $x$ in $A$ such that $\map f x \in N_\epsilon$ whenever $x \in N_\delta$.


Continuous Everywhere

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


Then $f$ is everywhere continuous if and only if $f$ is continuous at every point in $\R$.


Continuity on a Subset of Domain

Let $A \subseteq \R$ be any subset of the real numbers.

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


Then $f$ is continuous on $A$ if and only if $f$ is continuous at every point of $A$.