Topology on Euclidean space

Question

Find the functions $f,g$ such that, $a$ be a limit point in the domain of $f$ with $\lim_{x\to a}f(x)=b$, $\lim_{y\to b}g(y)=c$ but $\lim_{x\to a}g(f(x) \neq b$

Answer 1

Consider :

$$f(x)=\cases{x\sin\left(\frac{1}{x}\right)\quad\mathrm{if}\,x\neq0\cr0\quad\mathrm{otherwise}}\quad\mathrm{and}\quad g(y)=\cases{0\quad\mathrm{if}\,y\neq0\cr1\quad\mathrm{otherwise}}$$

If we restrict both $f$ and $g$ to $\mathbb{R}-\{0\}$, we see that $\lim_{x\to0,x\neq0}f(x)=0$ and $\lim_{y\to0,y\neq0}g(y)=0$.

But $g\circ f$ doesn't have any limit at $0$.