Prove with the ( ε , δ ) limit proof that a function is continuous

So, I have to prove that the limit on $$(0,0)$$ for the following function exists (or not!), with the (ε,δ) limit proof :

if for every $$ϵ > 0$$ there exists a $$δ$$ such that, for all $$x ∈ D$$ , if $$0 < | x − c | < δ$$, then $$| f ( x ) − L | < ϵ$$

The function is:

$$f(x,y)=\begin{cases} \frac {\sin(xy)}{x} & \text{ if } x \neq 0 \\ y & \text{ if } x = 0\end{cases}$$

I already know that the limit, if it exists, should be $$0$$ by doing reiterated and directional limits, but I am stuck in the proof of the limit itself when trying to get a similar norm, which is how we prove it. This is how far I got:

$$||f(x)|| = \frac {|\sin(xy)|}{|x|} \leq \frac {1}{|x|} \leq ???$$

Any help is welcome

• you are using wrong estimate. Instead note that around $0$, $|sin(x)| \le |x|$ – user160738 Apr 17 '19 at 9:05

For each $$t \in \mathbb R$$ we have $$|\sin(t) \le |t|.$$ Hence
$$|f(x,y)| \le |y|$$ for all $$(x,y)$$.
$$\delta =\epsilon$$ will do because $$|f(x,y)| \leq \max \{|x|,|y|\}$$ (since $$|sin\, t| \leq |t|$$).