TITLE: Is a multivariable function continuous iff it is continuous with respect to each variable?
QUESTION [5 upvotes]: I am very uncertain when it comes to understanding the continuity of multivariable functions.
If we have, for example, a function $f: \mathbb{R}^{4} \to \mathbb{R}$, and we denote the four variables $x,y,z,w$, are the following statements equivalent?

i) $f$ is continuous
ii) $f|_{x}, f|_{y}, f|_{z}, f|_{w}$ are each continuous

Here, $f|_{x}$ stands for the function attained by fixing the variables $w,y,z$.  I hope this question makes sense.

REPLY [8 votes]: It is true that if $f$ is continuous, then it is continuous viewed as a function of each of its variables separately. But the converse is false. 
For example, let 
$$
g(x,y) = 
\begin{cases}
\frac{xy}{x^2 + y^2}, & \text{if $(x,y) \ne (0,0)$;} \\
0, & \text{if $(x,y) = (0,0)$.}
\end{cases}
$$
Then is $g$ is continuous everywhere except $(0,0)$, even though it is continuous viewed in terms of each of its variables separately. To see that it is discontinuous at $(0,0)$, consider its limit at $(0,0)$ along each of the lines $x=0$, $y=0$, and $y = x$.
If you require an example in four variables, write $f(x,y,z,w) = g(x,y)$.