Exercise 30

We know that if a function of a single variable is differentiable at a point, then that function is also continuous at that point. In this exercise we determine that the same property holds for functions of two variables. A function $f$ of the two variables $x$ and $y$ is continuous at a point $(x_0,y_0)$ in its domain if $$\begin{gathered} \lim_{(x,y) \to (x_0,y_0)} f(x,y) = f(x_0,y_0) \end{gathered}$$ or (letting $x=x_0+h$ and $y = y_0 + k$, $$\begin{gathered} \lim_{(h,k) \to (0,0)} f(x_0+h,y+k) = f(x_0,y_0). \end{gathered}$$ Show that if $f$ is differentiable at $(x_0,y_0)$, then $f$ is continuous at $(x_0,y_0)$. (Hint: Multiply both sides of the equality that comes from differentiability by $\lim_{(h,k) \to (0,0)} \sqrt{h^2+k^2}$.)