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Is there a way to write a function so that for any $c$ such that $a < c < b$, $f(c)$ is always the same?

For example, if you had an increasing function up until $0$ at which point the $f(x)$ is $0$ all the way until $10$ when the function starts decreasing again.

I am not looking for horizontal lines or piecewise functions.

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    How about a definition by cases, e.g. $$f(x)=\begin{cases} x &\text{if }x<0\\ 0&\text{if } x\geq 0\end{cases}$$2012-09-15
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    What is your goal? Why do you *think* you don't want to define your function piece-wise?2012-09-15
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    So, you want a function that has one piece doing one thing, another piece that is constant, and another piece that does another thing, but even though the pieces are very different, you don't want to define the pieces separately?2012-09-15
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    Why are piecewise functions out of the question? Perhaps smooth bump functions are what you are looking for.2012-09-15

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