I have to detemine if these functions are uniformly continuous on the given intervals

1) f(x) = ln X on (0,1)

my solution :

Let ε >0 , we can let δ=ε

by the mean value theorm says there is c belongs to (0,1) such that

ln(X)-ln(Y)\ x-y =1\c <= 1

thus, whenever x-y < δ

So , we have ln(x) - ln(y)<= x-y< δ=ε

so the function is uniformly continuous on the interval .

Is my solution right ???

2)x\tan x on the interval (- infinity, infinity)

my solution :

tan-1(x) = cotan(x), it's not uniformly continuous and becomes infinitely large at the points pik, k = 0,1,2,3, x/tan(x) and not uniformly continuous at given interval.

I do not know how can i prove this by the definition . can anybody help me???

3)f(x)= cos(lnx) on the interval (0,1)

my solution:

f(x) is continuous on the interval (0,1)

for any two points from the interval (0,1) we can write the following expression:

|f(x1) - f(x2)| = |cos(ln(x1) - cos(ln(x2)| = |- 1/2 sin (lnx1 + lnx2)/2 sin (lnx1 - lnx2)/2 | = = 1/2 |sin(ln(x1x2)/2) sin (ln(x1/x2)/2)| <= 1/2, so the assumption of uniformly continuity is performed at this interval.

Is my solution right ?

4)f(x)=x2 tan-1x [0, infinity)

My solution :

I think it is the same on the first question

it's not continuous and becomes infinitely large at the points pik, k = 0,1,2,3, x/tan(x) and not continuous at given interval.

5)ex

on the interval (0,infinity)

my solution :

it is not uniformly continuous function , but i do not know how i can prove this by the definition

Please can anybody help me to solve these questions ???