Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be a function. I need to prove/disprove the claim which says:
$f$ is continuous $\iff$ $f(x,\cdot), f(\cdot, y)\colon\mathbb{R}\to\mathbb{R}$ are continuous for every $y,x$ respectively.
I am pretty sure that this claim isn't true, so I'm going with disprove here.
My counter example was the function $f(x,y) = \cfrac{xy}{x^2 + y^2}$ which is continuous for $x$ alone and for $y$ alone, but $\lim_{(x,y)\to(0,0)}f(x,y)$ doesn't exist. Which means $f$ isn't continuous.
However, I'm not sure if what I'm doing is right.
Specific example for disproof: Let $x=y$, then $f(x,y)=\frac{1}{2}$ for all $x$ and $y$. Therefore $\lim_{x\to 0,y\to 0}=\frac{1}{2}$ in this direction, which differs from letting them $\to 0$ separately, where the limit is $0$.