let $\{f_n\}_{n=1}^{\infty}$be a sequence of continuous functions defined on $[0, 1]$. Assume that $f_n(x) \rightarrow f(x)$ for every $x ∈ [0, 1]$. Which of the following conditions imply that this convergence is uniform?
$a.$ The function f is continuous.
$b.$ $f_n(x) ↓ f(x)$ for every $x ∈ [0, 1].$
$c.$ The function $f$ is continuous and $f_n(x) ↓ f(x)$ for every $x ∈ [0, 1].$
i thinks all option a) , b) and c) are true take $f(x) = -e^x$ strictly decreasing monotonics
Is its correct ???
No. Only c) is correct; it follows from Dini's theorem.
Option a) is false. Just take $f\colon[0,1]\longrightarrow\mathbb R$ as the null function and$$f_n(x)=\begin{cases}x&\text{ if }x=\frac1n\\0&\text{ otherwise.}\end{cases}$$And option b) is false too. Take $f_n\colon[0,1]\longrightarrow\mathbb R$ defined by $f_n(x)=x^n$. It converges pointwise (but not uniformly) to$$f(x)=\begin{cases}0&\text{ if }x<1\\1&\text{ otherwise.}\end{cases}$$