Continuity

A function $f$ is continuous at $a$ iff:

\lim_{x\to a}{f(x)}=f(a)

\forall{\epsilon>0}\; \exists{\delta>0}\; \forall{x}\; (|x-a|<\delta\implies{|f(x)-f(a)|<\epsilon})

One-Side Continuous

A function $f$ is continuous from left at $a$ iff:

\lim_{x\to a^{-}}{f(x)}=f(a)

A function $f$ is continuous from right at $a$ iff:

\lim_{x\to a^{+}}{f(x)}=f(a)

A function $f$ is continuous in $(a,b)$ iff $f$ is continuous on every $c\in(a,b)$ .

A function $f$ is continuous in $[a,b]$ iff $f$ is:

Suppose a function $f$ is continuous on $(a,b)$ . $f$ is uniformly continuous on $(a,b)$ iff:

\forall \epsilon >0\;\exists \delta >0\;\text{s.t.}\; |x-y|<\delta \implies |f(x)-f(y)|<\epsilon

If a function $f$ is continuous on $[a,b]$ , $f$ is uniformly continuous on $[a,b]$ .

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September 13, 2026 1 min read

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