sketch the graph of the function that is one-to-one on (+∞,-∞), yet not increasing on (+∞,-∞) and not decreasing on (+∞,-∞)

Question

sketch the graph of the function that is one-to-one on (+∞,-∞), yet not increasing on (+∞,-∞) and not decreasing on (+∞,-∞)

Answer 1

Every element in the domain maps to only one element in the range.
The derivative must be zero over all reals.
f'(x) > 0 means increasing and f'(x) < 0 means decreasing.
So the derivative must be zero over all reals.
This forces the function to be a constant function.
A constant function is typically a horizontal line.
BUT... nothing is said about continuity.
----------------------------------------------------------------------------------------------
How about this:

f(x) =

q/p if x is rational, x = p/q for integers p and q such that gcf(p,q)=1, p>0

or
0 if x=0

or

x if x is irrational
--------------------------------------------------
That is, is x is rational, change it to a fraction in lowest terms, flip it over, and plot it

Proof:

Let x1 and x2 be rationals
where x1 = p1/q1 and x2 = p2/q2 for integers p1,p2,q1,q2, p1>0, p2>0

f(x1) = f(x2) implies

f(p1/q1) = f(p2/q2)

q1/p1 = q2/p2

1/x1 = 1/x2

x2 = x1
That proves 0ne to one if x is rational.

If x is irrational, then f(x)= x which is the one to one identity line.

Moreover, the derivative is always zero.