Answer
Discontinuity can be removed.
Work Step by Step
We have to find the limit
\[
\begin{array}{ll}
f(x, y)=x^{2}+7 y^{2} & (x, y)=(0,0) \\
f(x, y)= & -4 \quad(x, y) \neq(0,0) \\
& =\lim _{(x, y) \rightarrow(0,0)} f(x, y) \\
& =\lim _{(x, y) \rightarrow(0,0)} x^{2}+7 y^{2} \\
& =0
\end{array}
\]
The limit does exit, so the discontinuity $f(0,0)=-4 \neq 0$ can be removed.
