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.