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

[Ivars]

@Ivars: this proof also works for hyperreals.

Based on "language analogy" between real and hyperreal sets?

I will look into this deeper before making any new statements.

At least, as a first step, is it possible to construct such discontinuos function f(x) on real numbers which in each point \( x_o \) is 0 if the point is approached from left, and 1 if approached from right( or vice versa)? So:

\( \lim_{x\to\\{-x_o}} f(x_o)= 1, \)
\( \lim_{x\to\\{+x_o}} f(x_o)= 0, \)

Alternatively :

\( \lim_{x\to\\{-x_o}} f(x_o)= 0, \)
\( \lim_{x\to\\{+x_o}} f(x_o)= 1, \)

Can we define a discontinuous function in such manner?

The one use of such function as a function of \( x_o \) would be for any \( x_o \) to discern from which side we are approaching it.

If we approach \( x_o \) from left, this function \( f(x_o) \) has value e.g. 0 (alternatively 1)
if we approach \( x_o \) from right ,this function \( f(x_o) \) has value e.g. 1 (alternatively 0)

Ivars