Question: \"Suppose W is a Path connected region, that is, given any two points

Question

Question: "Suppose W is a Path connected region, that is, given any two points of W there is a continuious path joining them. If f is a continuous function of, use the intermediate-value theorem to show that there is at least one point in W at which the value of f is equal to the average value ovver W, that is, the intergral of f over W divided by the volume of W. (compare this with the mean-value theorem for double integrals.) what happens if W is not connected?"

Hint: My professor gave us a hint showing the figure below. He said something if the path over W goes below and above the average value then the problem can be treated with the 2D verson of the intermediate-value theorem, but I was still a little confused and was wondering if someone could explain it a little more.

Explanation / Answer

W is a Path connected region, that is, given any two points of W there is a continuious path joining them.

From intermediate value problem, we know that, if f is a continuous function on the closed interval [a, b], and if d is between f(a) and f(b), then there is a number c [a, b] with f(c) = d.

Here, w is a closed path and f is a continuous function of w, then the average value over w is a point which exists on the w path i.e. f(w) exists.

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