A sufficient condition for differentiability for many variables

From a graphical perspective, when the derivative of a single variable function is continuous at a point, we can safely assume that the function is differentiable there. This is a mathematical way of defining the "smoothness" of a function. The same concept can be extended to multivariable functions. If the partial derivatives are continuous, the function is differentiable. This theory is a shortcut, a tool, to be used when we want to find whether a function is differentiable or not without the definition.

One may have asked about the directional derivative. Remember, the directional derivative finds a rate of change with a certain direction, not a function. It's meaningless to study continuity of values. We study continuity of functions.

Careful with some assumptions here! From a previous theorem we know that being differentiable implies in being continuous, because if the function is not continuous it cannot be differentiable. However, certain continuous functions fail to be differentiable. In addition, the partial derivatives themselves are not sufficient to prove that the function is differentiable. The fact that the function is differentiable does not imply that the partial derivatives are continuous. The function must be continuous to be differentiable, but the derivatives themselves may be discontinuous.