contraction?

Question

Is there anyone who can give me an example of a continuous function f: R-->R such that |f(x) − f(y)| < |x − y| for all x and y in R (x not equal to y) but such that f has no fixed point.

Answer 1

Imagine an increasing, concave up function with y=0 a horizontal asymptote to the left and y=x an oblique asymptote to the right and such that |f'(x)|<1 for all x. This can be done with a rotated hyperbola, for example. This will satisfy your conditions

Answer 2

f(x) = {2 if x≤1, x+1/x if x>1}

Note that this function is not only continuous, but also differentiable.

Edit: Paritosh Vasava, your function has a fairly obvious fixed point near 0.27846454276107.

Mathematician, what's with the down vote? The function I gave fulfills the requirements.

Answer 3

f(x) = e^-(x+1) has no fixed point on the reals, since x is never equal to e^-(x+1) for any real number.

And,
|f(x) − f(y)| < |x − y| for all x and y in R (x not equal to y) .

Answer 4

I'm not so sure but I think here's my answer: R-->R |f(x)-− f(y)| < |x − y| to X and y = N