Derivative Existence and Continuity: Unraveling the Mean Value Theorem

In summary, the conversation discusses Theorem 2-8 in "Calculus on Manifolds" by Spivak, which uses the Mean Value theorem to establish the existence of the Derivative assuming the existence of the partial derivatives. It is noted that this also assumes the continuity of the function, but the subsequent exercises show that the partial derivative may exist even if the function is not continuous. The speaker asks for clarification on this point. Ultimately, it is clarified that the mean value theorem is applied to the partial function g(x) = f(x, a2, ..., an), which is continuous because D1f = Dg exists, rather than to f itself.
  • #1
krcmd1
62
0
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
 
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  • #2
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. :)
 
  • #3
Thank you!
 

What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is defined as the slope of the tangent line to the function at that point.

Why is the existence of a derivative important?

The existence of a derivative is important because it allows us to analyze the behavior of a function and make predictions about its future values. It also helps us to find maximum and minimum points, as well as the direction of the function's growth or decline.

What is the formal proof for the existence of a derivative?

The formal proof for the existence of a derivative is based on the definition of a limit. It involves taking two points on the function that are very close together and finding the slope of the line that connects them. As the distance between the two points approaches zero, the slope of this line will converge to a specific value, which is the derivative at that point.

Can the existence of a derivative be proven for all functions?

No, the existence of a derivative cannot be proven for all functions. Some functions may have points where the derivative does not exist, such as sharp corners or discontinuities. However, for most continuous functions, the derivative can be proven to exist at all points within its domain.

How is the existence of a derivative related to the continuity of a function?

The existence of a derivative is closely related to the continuity of a function. A function is said to be continuous at a point if its limit exists at that point and is equal to the value of the function at that point. Similarly, the existence of a derivative at a point means that the function is continuous at that point.

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