Total differentiability implies continuity
If a multivariate function is totally differentiable, then it is continuous. But the converse is not true. The existence of partial derivatives is insufficient for a function to be totally differentiable. However, if partial derivatives exist AND they are continuous, then the function is totally differentiable. Just like the univariate case, we can also define the notion of differentiability classes such as class C^1 or continuously differentiable functions.
Lecture notes: https://bruneimathclub.blogspot.com/2...
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