# Function with f(0)=0,f'(0)=+inf,f(inf)=<1

f(x) should give the chance of something happening, thats the reason for f(inf)=<1.

i used f(x)=1-exp(-a*x) until now, which is ok but f'(0) is not +inf. i would prefer a function with f(0)=+inf.

i want the function to be (monotonically) increasing at a decreasing rate (f' decreasing monotonically from +inf to 0)

does a function like this exist?

mathman
Play around with xlnx.

Mentallic
Homework Helper
When you said $f( \infty )\leq 1$ are you sure that's what you mean? So you could have $f(\infty)=-\infty$?

i dont see how i can get what i want from x ln(x)... can you eloborate? i mean x ln (x) goes to infinity

i meant i want f(inf)=1 (monotonically increasing from 0 to inf.), so f'(inf)=0.

chiro
f(x) should give the chance of something happening, thats the reason for f(inf)=<1.

i used f(x)=1-exp(-a*x) until now, which is ok but f'(0) is not +inf. i would prefer a function with f(0)=+inf.

i want the function to be (monotonically) increasing at a decreasing rate (f' decreasing monotonically from +inf to 0)

does a function like this exist?

Look up the heaviside and dirac delta functions and use these to create your function.

i was shocked at the functions you came up with... then is saw my typo:

i would prefer f'(0)=inf (and not f(0)=inf, since i want f(0)=0)

mathman
When I said play around with x lnx, I was thinking of something like (x lnx)/(1 + x2)

Look up the heaviside and dirac delta functions and use these to create your function.
Uuuh, these are distributions. I doubt the OP wants those.

Char. Limit
Gold Member
What about log(x-1)/(x-1) + i pi?

EDIT: Never mind this. It assumed f'(1)=+inf, not f'(0)=+inf.

x^(1/3) - (x-1)^(1/3) + 1 should work though.

Last edited:
a bit late, but thanks all for your efforts!!

Char Limit's thing works perfectly!! :-) ... after changing the signs:

-x^(1/3)+(x-1)^(1/3)+1

f(x) = sqrt(x) / (1+sqrt(x))
more general : f(x) = (x^a)/(1+(x^a)) with 0<a<1

hmmm, at first i thought -x^(1/3)+(x-1)^(1/3)+1 worked (i tested only for integers) but f is not monotonically increasing (and f' is not monotically decreasing)... and worse, f even gets negative...

but jacquelin your solutions does work perfectly! :-) im very happy with your quick response! thanks alot!