This is an extremely open question; there are very many answers. You might consider polynomials, or rational functions, or exponentials, or logarithms, or many other categories to look for something that is one-to-one.
Not really wanting to hijack this thread but I have a question. 1st, if the question has many replies, then can it please be moved to its own thread. After reading through counterexamples in analysis by Gelbaum(?) I think that anything can be true. Is there a function that is not 1-1 and no portion of it is 1-1?
For example y=sin(x) is not 1-1 but we can find a portion that is 1-1.
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