Derivative Existence and Continuity: Unraveling the Mean Value Theorem

[krcmd1]

I have been trying to teach myself math, and for quite a while have been struggling through "Calculus on Manifolds" by Spivak.

Theorem 2-8, on p.31, uses the Mean Value theorem to establish the existence of the Derivative assuming the existence of the partial derivatives.

Doesn't that also assume the continuity of the function? If I've understood the subsequent exercises, the partial derivative may exist even though the function may not be continuous.

What am I missing?

Thanks very much, again!

Ken Cohen

 

[slider142]

He applies the mean value theorem to the partial function g(x) = f(x, a², ..., aⁿ), which is continuous because D₁f = Dg exists. He does not apply it to f. :)

 

[krcmd1]

Thank you!