Exercises for Section 10.2 A. Assuming that the Weierstrass Theorem is true for C[0, 1], prove that it is true for C[a, b], for an arbitrary interval [a, b]. HINT: For $f \in C[a, b]$, use $g(t) := f(a + (b - a)t)$ in C[0, 1].
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This means that every continuous function on the interval $[0, 1]$ attains its maximum and minimum values. Show more…
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(a) Prove that if $f$ is continuous on $[a, b],$ then there is a function $g$ which is continuous on $\mathbf{R},$ and which satisfies $g(x)=f(x)$ for all $x$ in $[a, b]$ Hint: since you obviously have a great deal of choice, try making $g$ constant on $(-\infty, a]$ and $[b, \infty)$ (b) Give an example to show that this assertion is false if $[a, b]$ is replaced by $(a, b)$
(a) Assume that $g$ and $h$ are continuous on $[a, b]$. Use Corollary 2 to show that if $g(a)<h(a)$ and $h(b)<g(b),$ then there exists $c \in[a, b]$ such that $g(c)=h(c)$ (b) Interpret the result of (a) in terms of the graphs of $g$ and $h,$ and show, by a graphical example, that the conclusion in (a) need not hold if one of $g$ or $h$ is not continuous.
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