Math Problems: Real Analysis

Page 1 Page 2 Page 3 Problems Home

Problem 1

1. Show that whenever (a_n) and (b_n) are two sequences of real numbers where b_n is positive, strictly increasing, unbounded, and limit((a_(n+1)-a_n)/(b_(n+1)-b_n),+oo)=l < +oo, then limit(a_n/b_n,+oo) also exists and is equal to $l$.
Solution

Problem 2

2. Let f: R -> R be a function with graph G(f) such that for every point (a,0) there exists a point A on G(f) such that d((a,0),A) = d((a,0),G(f)). We call this distance g(a). Prove that for all a,b in R,|g(a)-g(b)| <= |a-b|.
Solution

Problem 3

3. f: [a,b] -> R is such that for every epsilon >0, there is d>0, d>sum(|b_k-a_k|,k=1..n) => epsilon>sum(|f(b_k)-f(a_k)|,k=1..n) (in particular, f is absolutely continuous). Show that f is lipschitz.
Solution

Problem 4

4. Let f_n be a family of real increasing functions defined over an open interval such that the sequence $f_n(x)$ is bounded for every x. Show that there exists a subsequence k_n such that f_(k_n)(x) converges for every x.
Solution

Problem 5

5. Give a bijective function f continuous at 0 such that its inverse is not continuous at f(0).
Solution

Problem 6

6. Assume f: R -> R admits primitives, and f^2(x)=x^2. Show that f(x)=x or f(x)=-x or f(x)=|x| or f(x)=-|x|.
Solution

Problem 7

7. Find a function f: R -> R which is measurable, satisfies the intermediate values property on every interval of R, and does not admit any primitive on R.
Solution

Problem 8

8. Let f be differentiable on [a,b], and such that for every x in ]a,b[, f'(x)>(f(x)-f(a))/(x-a). Prove that f'(b)>f'(a).
Solution

Problem 9

9. Find all functions f:[0,1] -> [0,1] such that for every x,y in [0,1],|f(x)-f(y)| >= |x-y|.
Solution

Problem 10