Find its pointwise limit and determine if its uniform

Question

$f_k:[0,1]\to\Bbb R,$ $f_k(x)=\begin{cases}kx,x\in[0,\frac 1k]\\1,x\in[\frac1k,1]\end{cases}$

Find the pointwise limit of $(f_k(x)) $. And is this convergence uniform?

for x=0 $f_k(0)=0$ and x=1 $f_k(1)=1$

How can I start?

Answer 1

If $k\to \infty$, $\frac1k\to 0$ so the limit of $f_k(x)$ is $$f(x)=\begin{cases}0,x=0\\1,x\neq 0\end{cases}$$

For the convergence uniform, it can not be since $f$ is discontinuous on $0$.