Chapter 13 - Partial Derivatives - 13.4 Differentiability, Differentials, And Local Linearity - Exercises Set 13.4 - Page 947: 29

$\underline{True}$.

According to theorem 13.4.4, we know that if all the first order partial derivatives of a function $f$ exist and are continuous at a point, then the function $f$ is differentiable at that point, so the function should be continuous.