The following was posted very long time ago, I am posting it here. Let kill them :

Deal All, I have some more problems which need your kind help:

1) f(x)=3x^2 when x is irrational but -5x^2 when x is rational, ask f(0) is continuous? differentiable?

I think it is both continuous and differentiable at 0, but not sure the reasons

2) A similar one as above, g(x) = 1 when x is rational and e^x when x is irrational, the answer is g(x) is continuous only at 0. Why?

3) f’(0)=f’’(0)=1, g=f(x^10), ask the 11-th differentiation of g at 0.

4) When x>=0, x-x^3<=f(x)<=x, f(0) is differentiable?

5) Ring R has the following properties but R*R (pointwise) doesn’t have:

a. R is field

b. R is finite

c. R is communicative

6) find out the number of connected component of e^z where |z|=1

7) f: X->Y is continuous bijection, which are correct?

a. if X is compact , so is Y

b. if X is Hausdorff space, so is Y

c. if X is compact and Y is Hausdorff space , then inverse of f exist

Cool f’’(0) < 0 and f’(0) = 0, let T=f(0)+2f(2)+2f(4)+2f(6); I=integration of f(x) from 0 to 6; R=2f(2)+2f(4)+2f(6); sort T, I and R

9) Fair coins are tossed and when either four consecutive heads of tails appear the process stops. What is the probability of two consecutive head or tail or any one of them in one row?

#2 was solved already

I start with # 5 ( the easiest one)

5) Ring R has the following properties but R*R (pointwise) doesn’t have:

a. R is field

b. R is finite

c. R is communicative

I have no idea with this question