I need to study the continuity of the following function: $f(t,x)=\left\{\begin{matrix}0, \ \ \ \ \ t \le 0 \\ 2t, \ \ \ \ \ t > 0, y < 0 \\ 2t − \frac{4y}{t}, \ \ \ \ \ t > 0, 0 \le y \le t^2 \\ -2t, \ \ \ \ \ t > 0, t^2 < y \end{matrix}\right.$
What I did:
Let $S_i, i=0,1,2,3, $ be the 4 domains of the piecewise definition of $f$.
We have $\mathbb{R}^2 = S_0 \cup S_1 \cup S_2 \cup S_3 $
$f$ is clearly continuous in the 4 domains as the sum, product or quotient of continuous functions (the denominator not vanishing to 0).
Remains to study the continuity at the frontiers of the 4 domains.
I found the following cases to study but I don't know if I am missing something:
We have 4 domains, each 2 domains have common frontier so the number of limits to study is $C_4^2=6$
$0=\lim_{\substack t\rightarrow 0 \\ t<0}f(t,y)=\lim_{\substack t\rightarrow 0\\ t>0\\ y <0 }f(t,y)=\lim_{\substack t\rightarrow 0\\ t>0\\ y <0 }2t=0$
$0=\lim_{\substack t\rightarrow 0 \\ t<0}f(t,y)=\lim_{\substack t\rightarrow 0\\ t>0\\ 0 \le y \le t^2 }f(t,y)=0$
$0=\lim_{\substack t\rightarrow 0 \\ t<0}f(t,y)=\lim_{\substack t\rightarrow 0\\ t>0\\ y > t^2 }f(t,y)=\lim_{\substack t\rightarrow 0\\ t>0\\ y > t^2 }-2t=0$
$0=\lim_{\substack y\rightarrow 0 \\ y<0 \\t>0}f(t,y)=\lim_{\substack y\rightarrow 0\\ t>0\\y>0 \\ y \le t^2 }f(t,y)=0$
$t^2=\lim_{\substack y\rightarrow t^2 \\ t>0 \\y \le t^2}f(t,y)=\lim_{\substack y\rightarrow t^2\\ t>0\\y>t^2}f(t,y)=t^2$
$0=\lim_{\substack t\rightarrow 0 \\ t<0}f(t,y)=\lim_{\substack t\rightarrow 0\\ t>0\\ y > t^2 }f(t,y)=0$
Is there a more concise way to study the continuity of such functions ? Any help much appreciated