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I'm trying to find a function of single variable f(x) with the following properties:

-It is symmetric around zero

-It is differentiable everywhere

-f'(x)≥0 for all x>0

-f'(x)=0 when x=0

-f'(x)≤0 for all x<0

(I think these last two actually follow from the first three?)

-It has an upper bound of 1

-It has a lower bound between 0 and 1, which I can set using a parameter c

For example, the function

f(x)=(c/10)*abs(x)+1-c when abs(x)≤10

f(x)=1 when abs(x)>10

(0≤c≤1)

fulfills all the conditions, except that it is not differentiable at x=-10, x=0 and x=10.

But I'm hoping to find a relatively simple function that would fulfill all those requirements.

If anyone has any ideas, I'd be very thankful!