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 ???