Answer
This is true.
Work Step by Step
This is true.
Since $g$ and $h$ are continuous then $$\lim_{x\to x_0}g(x)=g(x_0)$$ and
$$\lim_{y\to y_0}h(y)=h(y_0).$$
We have
$$\lim_{(x,y)\to(x_0,y_0)}f(x,y)=\lim_{(x,y)\to(x_0,y_0)}(g(x)+h(y)) =\lim_{(x,y)\to(x_0,y_0)}g(x) + \lim_{(x,y)\to(x_0,y_0)} h(y) = \lim_{x\to x_0}g(x)+\lim_{y\to y_0}h(y) = g(x_0)+h(y_0)=f(x_0,y_0),$$
which means that $f$ is continuous.
