Show f need not be continuous.

Question

If the functions $f_k$ are lower semicontinuous, $f_k \rightarrow f$ pointwise, and $f_{k+1} \geq f_k(x)$ then $f$ is lower semicontinuous. Show that $f$ need not be continuous even if the $f_k$ are continuous.

I proved the first statement by showing if the hypothesis hold then, $\lim_{x\rightarrow x_0}inf f(x) \geq \lim_{x\rightarrow x_0}inf f_k(x) \geq f_k(x_0) $.

Answer 1

For example, if we take $f_k:[0,1] \to \Bbb R$, try the sequence $f_k(x) = 1 - x^k$. What is its pointwise limit?