Exercise 45

Prove that if \(f\) is a function of two variables that is differentiable at \((a, b)\), then \(f\) is continuous at \((a, b)\).

Hint: Show that
\[ \lim_{(\Delta x, \Delta y) \to (0, 0)} f(a + \Delta x,\ b + \Delta y) = f(a, b) \]