Answer
Not true.
Work Step by Step
This not true. The counterexample is
$$f(x,y)=\left\{\begin{array}{cc}
\frac{\sin xy}{xy},&(x,y)\neq(0,0);\\
0,&(x,y)=(0,0).
\end{array}\right.$$
This function is continuous in every point when $x$ and $y$ are nonzero as a quotient of continuous functions $\sin xy$ and $xy$ but it limit is
$$\lim_{(x,y)\to(0,0)}\frac{\sin xy}{xy} =1$$
which is different than $f(0,0)=0.$
