If $f_m$ is a sequence of continuous functions which converges uniformly to a function. then that function is continuous as well.

Question

I was looking for the proof is this theorem, but I couldn't find it anywhere.

the theorem is stated formally:

If $f_m$ is a sequence of continuous functions defined on $D$ (subset of $R$) such that $f_m$$\to$$f$ uniformly on $D$ then $f$ is continuous.

can someone give the stepwise proof?

Note that uniform convergence is critical here. Without it, the limit need not be continuous (consider $f_n(x)=x^n$ on $[0,1]$). — MPW, Nov 25 '18 at 17:10

score 0 · Answer 1 · answered Nov 25 '18 at 17:13

0

Hint

To prove that $f$ is continuous at $a\in D$,

use the fact that, for great enough $n$, and $x\in D$,

$$|f(x)-f(a)|\le $$ $$|f_n(x)-f(x)|+|f_n(x)-f(a)|$$

answered Nov 25 '18 at 17:13

hamam_Abdallah

1 Answers1