The function is continuous for all real x for any choice of a.
(I tell my algebra students the domain of a function is all real numbers... unless it isn't.)
The function is smooth everywhere except at x=0. Think of the function without the absolute values around the x's. f(x) is just the values with positive x reflected over the y-axis. To make it smooth at x=0, we need the left and right hand derivatives of f(x) at x=0 to be equal. By symmetry, they will both need to equal zero.
I'll spare the details.
For the left replace |x| with -x, take the derivative and substitute x=0 to get -2a(5a-1)e^25
For the right replace |x| with x, take the derivative and substitute x=0 to get 2a(5a-1)e^25
These are both equal if a=0 or a=1/5
i) a is any real number
ii) a is 0 or 1/5
Posted by Jer
on 2021-11-27 09:40:22