MT2002 Analysis

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## Exercises 10

1. If f, g are continuous functions on an interval [a, b], prove that d1(f, g) ≤ (b - a)d(f, g).
Hence prove that if a sequence (fn) in C[a, b] converges to a function f in d it also converges to f in d1.

2. Let f be a continuous function on the interval [0,1] for which f (0) = f (1). By considering the function h(x) = f (x) - f (x + 1/2) on a suitable interval, prove that there is some point x in [0, 1/2] for which f (x) = f (x + 1/2).
Deduce that there are two points opposite one another on the equator where the temperature is the same.

3. Let f be a function from R to R for which there is a constant λ < 1 such that | f (x) - f (y) | ≤ λ| x - y|. Such a function is called a contraction mapping.
Choose a point x0 and define a sequence (an) by a1= x0, an+1= f (an) for n ≥ 1. Prove that this is a Cauchy sequence and that it has a limit a which is a fixed point of f. i.e. f (a) = a.
Prove that this fixed point is unique.

The next three questions try to show you why the metric d2 is an easier one to work with than d1 or d.

4. Find the best-fit straight-line through the origin to the function x2 in the metric d1.
i.e find the value of a which minimises d1(x2, ax).
For this you have to minimise the shaded area.
Calculate this area by first finding the x-coordinate of the point P in terms of a.

5. Find the best-fit straight-line through the origin to the function x2 in the metric d.
i.e. find the value of a which minimises d(x2, ax).
For this you have to choose a so that the "vertical distance between the graphs" is a minimum.
From the picture, you can see that the best you can do is to adjust the value of a so that the two dark lines have equal length.

6. Use the method of Exercises 9 Question 1 to find the best-fit straight-line through the origin to the function x2 in the metric d2.

SOLUTIONS TO WHOLE SET
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JOC September 2001