Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 8
Exercise 1 Let f:IโR, where Iis a real interval, be a function. We say that fis C-Lipschitz on I,
where C > 0, if for every x, y โIwe have |f(x)โf(y)| โค C|xโy|.
(1) Assume fis C-Lipschitz. Prove that fis continuous.
(2) Prove that the function x7โ |x|is 1-Lipschitz.
(3) Prove that the function x7โ x2is 2-Lipschitz on [0,1].
(4) Prove that the function x7โ x2is C-Lipschitz on any compact interval.
(5) Prove that the function x7โ x2is not C-Lipschitz on Rfor any C.
Exercise 2 Let f:RโRbe a continuous function. We assume that limxโ+โf(x) =
limxโโโ f(x) = +โ. Prove that there is x0โRsuch that f(x0)โคf(x)for every xโR.
Exercise 3
(1) Let f:RโRbe a continuous function. Prove that its graph Gf:= {(x, y)โR2, y =f(x)}is
closed.
(2) Let f:RโRbe the function defined by f(x) = 1
xif x6= 0 and by f(0) = 0. Prove that its graph
is closed. Find an open set in Rsuch that its inverse image by fis not open.
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