Math Is Fun Forum

dchilow

Member

Registered: 2007-03-05

Posts: 27

Help with Real analysis.

1)  Use (For all epsilon>0)(There exists delta>0) such that (0<|x-c|<delta ⇒ |f(x)-L|<epsilon) to prove limx->c (f(x))=L for the following:

(a)  limx->4 (√(x))=2

(b)  limx->2 ((x³-8)/(x²-4))=3

2)  Let I be an open interval that contains the point c and suppose that f is a function that is defined on I except possibly at the point c.
            (a)  State the definition for a function that does not have a limit at c.  Then use it to prove that limx->0 (sin(1/x)) does not exist.  (Hint:  limx->c (f(x))=L if and only if limn->infinity (f(Xn))=L)

            (b)  State the definition for a function that does not have a limit at any point on the interval I.  Then use it to prove that the function X(CHI)subQ (x) does not have limit at any point.

3) Let f: (a,b) -> R be continuous at c ∈(a,b) and suppose that f(c)>0.  Prove that there exist an interval (u,v)C(a,b) such that c ∈ (u,v) and f(x)>0 for all x ∈ (u,v).

4) Suppose that f is a continuous function defined on an interval I. Prove that |f| is continuous on I.

5)  Suppose f is an increasing function defined on an interval I.  Prove that f can have at most countably many discontinuities. 

Please, this is my practice test to get ready for my final exam, could someone please help me out on these, I would appreciate it very much.