Continuity of Multivariable Piecewise Functions

Question

Let's say we have a function

$$f(a,b) = \begin{cases} a &,ab \geq 0\\0 &,ab < 0\end{cases}$$

Is this function going to be continous everywhere since the limit of $f(a,b)$ equals $f(a,b)$ everywhere, or is there an exception when $ab = 0$?