In single variable calculus, a differentiable function is necessarily continuous (and thus conversely a discontinuous function is not differentiable). In multivariable calculus, you might expect a similar relationship with partial derivatives and continuity, but it turns out this is not the case! In this example, what I call the cross function, we will see that both partial derivatives exist but that the function is nonetheless not continuous. This means our work is still cut out for us: we need to define a new concept to replace single variable differentiation in the multivariable context that is more than just partial derivatives existing.


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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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