Is the limit of an increasing sequence of continuous functions is a lower semicontinuous function

Question

1.$ f_n(x)$ is continuous，bounded，positive and $f_n(x)\le f_{n+1}(x)$

2.$ f(x)=\lim f_n(x)$

Question：Is $f(x)$ a lower semicontinuous function？

Answer 1

Consider, $(f_n)\subset C[0, 1]$ defined by $$f_n(x) =x^n$$

Then, $f_n \to f$ pointwise on $[0, 1]$

where, $f: [0, 1]\to \mathbb{R}$ defined by $$f(x) = \begin{cases} 0 & 0\le x <1 \\ 1 & x=1 \end{cases}$$

Here, the function $f$ is not lower semi continuous.

\begin{align}m_f{(1)}&= \lim_{\delta \to 0} \space inf\{f(x): x\in (1-\delta,1+\delta) \} \\&=0 \end{align}

Hence, $m_f{(1)}\neq f(1) $

Hence, $f$ is not lower semi continuous at $1$