Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp

[bo198214]

No, for all real points \( x_0 \).

Such a function can not exist. If \( \lim_{x\uparrow x_0}f(x)=0 \) then there must exist a \( \delta>0 \) such that \( f(x)=0 \) for all \( x_0-\delta<x<x_0 \), because otherwise - if in each left neighborhood of \( x_0 \) there is an \( x \) with \( f(x)=1 \) - there is a sequence \( x_n\to x \) with \( f(x_n)=1 \) which means that \( \lim_{x\uparrow x_0}f(x) \) does not exist or is 1.
But if \( f(x)=0 \) in whole left neighborhood of \( x_0 \) then \( \lim_{x\downarrow x_1} f(x) = 0 \) for \( x_1 := x_0 - \frac{\delta}{2} \).