question related to equicontinuous set

Question

I have come across one problem. I have a set, $T=\{f | f:[a, b]\rightarrow \mathbb{R}\}$ where each $f$ is a piecewise continuous function on $[a, b]$. Can we talk about equicontinuity of the ser $T?$

Answer 1

No.

If $F$ is a family of equicontinuous functions, then each $f \in F$ is continuous. Your Family $T$ contains many functions, which ar not continuous.

Answer 2

(1).Yes. This comes up in the proofs of existence of solutions of certain integral or differential equations. Let $F$ be a family of continuous functions from $[a,b]$ to $\mathbb R.$ For $x\in [a,b]$ we say that $F$ is equicontinuous at $x$ iff $$\forall r>0 \;\exists s>0 \; \forall f\in F \;(y\in [a,b]\land |y-x|<s\implies |f(y)-f(x)|<r).$$ (2).A continuous $f:[a,b]\to \mathbb R$ is uniformly continuous. That is $$\forall r>0\;\exists s>0\; ((x,y\in [a,b]\land |x-y|<s)\implies |f(x)-f(y)|<r).$$

If the $F,$ above, is equicontinuous at every $x\in [a,b]$ then we also have $$\forall r>0\; \exists s>0\; \forall f\in F \;((x,y\in [a,b]\land |x-y|<s)\implies |f(x)-f(y)|<r).$$ To prove this, by contradiction, suppose not. Then take $r_0>0$ such that for each $n\in \mathbb N$ we have some $f_n\in F$ and $x_n,y_n \in [a,b]$ with $$(0<|x_n-y_n|<1/n)\; \land \;(|f_n(x_n)-f_n(y_n)|\geq r_0).$$ Now let $(x_{n_j})_j$ be a convergent subsequence of $(x_n)_n,$ converging to $x.$ Since $|x_{n_j}-y_{n_j}|<1/n_j,$ the sequence $(y_{n_j})_j$ also converges to $x.$

Now $F$ is equicontinuous at $x,$ so take $s>0$ such that $$\forall f\in F ((y\in [a,b]\land |y-x|<s)\implies |f(y)-f(x)|<r_0/2.$$ But there exists $j$ such that $\{x_{n_j},y_{n_j}\}\subset (x-s,x+s)$. (In fact this holds for all but finitely many values of $j$.) For such $j$ we have $$r_0\leq |f_{n_j}(x_{n_j})-f_{n_j}(y_{n_j})|\leq |f_{n_j}(x_{n_j})-f_{n_j}(x)|+ |f_{n_j}(x)-f_{n_j}(y_{n_j})|<r_0/2+r_0/2=r_0,$$ implying $r_0<r_0.$