I need to give explicit examples of two functions
$f$: [0,1]$\in\to\mathbb{R}$ and $g$: (1,2]$\in\to\mathbb{R}$ such that $f$ and $g$ are continuous,
but the function $h$: [0,2]$\in\mathbb{R}$ defined by $h(x)=$$\ \begin{cases}f(x)&x\in[0,1]\\g(x)&x\in(1,2]\end{cases}$.
Cannot use anything to do with limits. I could come up with functions for $f$ and $g$ that fit but I'm not sure if there is anything that will help me so it fits $h$ besides guess and check?
I think (if I understand your question) you just need $h$ to be discontinuous at $x=1$ (since it is clearly continuous on $[0,1]$ and $(1,2]$). In other words you need $f(1)\neq \lim_{x\to1^{+}}g(x)$. There are loads of options, eg. $f(x)=1$ and $g(x)=2$.