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

Continuous from left

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

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

Continuous from right

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

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

On interval

Open interval

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

Closed interval

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

• continuous on every $c\in(a,b)$
• right-continuous at $a$
• left-continuous at $b$

Uniformly continuous

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 $[a,b]$.