Uniform Convergence

Pointwise convergence

Pointwise Convergence

Let $S$ be a set, $(Y,\rho )$ a metric space, and $f:S\to Y$ a function. Consider a sequence of functions $(f_n)$ from $S$ to $Y$. We say $f_n\to f$ pointwise if for all $x\in S$ there holds $\rho (f_n(x),f(x))\to 0\text{as}n\to \infty$

Uniform Convergence

Let $S$ be a set, $(Y,\rho )$ a metric space, and $f:S\to Y$ a function. Consider a sequence of functions $(f_n)$ from $S$ to $Y$. we say $f_n\to f$ uniformly on $S$ if for any $\epsilon >0$ there exists $N\in \mathbb{N}$ such that for all $n>N$ there holds $\rho (f_n(x),f(x))<\epsilon \text{ for all }x\in S$Equivalently, ${\sup }_{x\in S}\rho (f_n(x),f(x))\to 0\text{as}n\to \infty$

Proposition

Uniform convergence implies pointwise convergence.

Theorem

Let $(X,d)$ and $(Y,\rho )$ be metric spaces, and let $f_n:X\to Y$ be a sequence of continuous functions such that $f_n\to f$ uniformly. Then $f$ is continuous.

Thrm Let $f_n:[a,b]\to \mathbb{R}$ be a sequence of continuous functions such that $f_n\to f$ uniformly on $[a,b]$ for some function $f:[a,b]\to \mathbb{R}$. Then ${\int }_a^b{f}_n(x)\text{dd}x\to {\int }_a^{b}f(x)\text{dd}x$Proof

Def Bounded Functions Let $(X,d)$ and $(Y,\rho )$ be metric spaces. Then