Math Problems: Real Analysis

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Problem 11

11. Show that the euclidean plane R^2 cannot be covered by disjoint circles (a point is not considered a circle).
Solution

Problem 12

12. Let f be a differentiable function on [0,1] such that f(0)=0 and f(1)=1. Show that there are two distinct points x,y such that f'(x)f'(y)=1.
Solution

Problem 13

13. Consider a norm N over the vector space R^n, such that for all (x_1,...,x_n), N(x_1,...,x_n)=N(|x_1|,...,|x_n|). Show that N(a_1+b_1,...,a_n+b_n) >= N(a_1,...,a_n) whenever a_i,b_i >= 0.
Solution

Problem 14

14. Is it true that the sequence (x_n) of real numbers is convergent if and only if limit(limsup_m |x_{n}-x_{m}|,n=+oo)=0?$
Solution

Problem 15

15. f is a function f: [0,1] -> R such that f(0)>0 and 0>f(1). Moreover, there exists a continuous function g: [0,1] -> R such that f+g is nondecreasing. Show that there is x in (0,1), f(x)=0.
Solution

Problem 16

16. Consider a function f:[0;1] -> R which is two times differentiable, and such that |f ''| >= 1 and ff ' <> 0. Show that there exists x in [0,1],|f(x)| >= 1/2}.
Solution

Problem 17

17. Let f be a function continuous on [0,1] such that f(1)=0. Show that there is some x_0 in (0,1) such that f(x_0)=int(f(u)du,u=0..x_0).
Solution

Problem 18

18. Show that there is no continuous function f: R -> R such that f(x)=0 admits exactly 2 solutions for each real c.
Solution

Problem 19

19. Given a convex real function f: R -> R such that f is C^2, show that int(f(x)cos(x)dx, x=0..2Pi) >= 0.
Solution

Problem 20