The function $f$ defined on the unit square $0 \leq s \leq 1$, $0 \leq t \leq 1$ is given below:
$$\begin{equation*} f(s,t) = \begin{cases} 1, & \text{if } s = t =0,\\ \frac{s+t}{\sqrt{s^2 + t^2}}. & \text{otherwise}. \end{cases} \end{equation*}$$
And the book said that this is a function of two variables that is continuous at every point of its domain of definition with respect to each variable but is not continuous in both variables simultaneously.
My question is:
Why the function is not continuous in $s$ and $t$ simultaneously?
I took the path $s =2t$ and I got the limit $3/\sqrt{5}$ at $(s,t) \rightarrow (0,0),$ but $f(0,0) =1$. Is this is the reason that the function is not continuous at $s$ & $t$ simultaneously?