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These Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson
Proofs of one-sided theorems could be of interest in the study of 2D topology. |
Intermediate Value TheoremTheorem F.3. [Intermediate Value Theorem] If a real-valued function f(x) is continuous at each point x in the interval [a,b] and (i) f(a) < y < f(b) or (ii) f(a) > y > f(b) then there exist at least one point x in the interval [a,b] such that f(x) = y. This theorem implies if y = l·f(a)+(1-)·f(b) for some real number l satisfying 0 £ l £ 1 then y = f(x) for some x in the interval [a,b] provide f(x) is continuous at all points in this interval.Proof of Intermediate Value Theorem. We only consider the case where f(a) < y < f(b). The other case f(a) > y > f(b) is similar. Now suppose f(a) < y < f(b)
and let
Then g(x) is a continuous, increasing function of x with g(a) = f(a) and g(b) ³ f(b) > f(a). To show g(x) is continuous at a given x0
in [a,b], let e
> 0. Now choose d > 0 such
that |x-x0|
£ d
implies |f(x)-f(x0)|
£ [1/2]e.
Put x1 = max(a,x0-d).
Then for |x-x0|
£ d,
But c Î [x1,x] Ì [x1,x0+d] implies |f(x1)-f(c)| £ |f(x1)-f(x0)+(f(x0)-f(c))| £ |f(x1)-f(x0)|+|f(x0)-f(c)| £ [1/2]e+[1/2]e = e. Therefore |f(x1)-f(c)| £ e and thus
£ g(x1)+e whenever |x-x0| £ d. The foregoing shows |x-x0| £ d implies |g(x)-g(x0)| £ e. Thus g(x) is continuous at each x0 in [a,b] If we show that y = g(x) for some x Î [a,b] then y is a limit point of the range of the continuous function f. The latter implies there exists at least one point x0 Î [a,b] such that f(x0) = y. (The point x0 could even be selected from the interval [a,x].) Let A = sup{x Î [a,b] | g(x) < y}. Then g(A) £ y and x > A implies g(x) ³ y. For the sake of contradiction suppose g(A) < y. Then g(x) has a jump of magnitude ³ y-g(A) at x = A (why?). The latter implies g(x) is not continuous at x = A. This contradicts the just demonstrated continuity of g(x) at all points x in the interval [a,b]. | ||||||||||||||||||||||||||||||||||||||||||||||||
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Road
Safety Message Do not walk on a road with your back to the
traffic - rule of thumb
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