In discussing how the concept of differentiability implying continuity cannot be applied to functions of several variables, Apostol proceeds to give an example to demonstrate why. The function he uses is as follows,
$f(x,y) = \left\{\begin{matrix} x+y,\textrm{if } x=0 \textrm{ or }y=0\\ 1, \textrm{otherwise} \end{matrix}\right.$
Apostol goes on to state that the partial derivatives at $(0,0)$ are both 1, but $f(x,y)$ at $(0,0)$ is not continuous. How is this so since $f(0,0)=0$?
The function $f(x,y)$ is not continuous at $(0,0)$, because if you approach this point from e.g. the line $y=x$, then you can consider the limit of $f(x,y)=f(x,x)$ as $x$ approaches $0$. We have that $f(x,x)=1$ for all $x\neq 0$, which is different from the value at $(0,0)$, which is $f(0,0)=0$.
The partial derivatives, however, exist at $(0,0)$, because these are derivatives taken along the axes, and $f$ is defined in a very specific way on the axes.