Assuming you are talking about non-piecewise linear functions, you might want to look at absolute values:
$f(x) = a \left | x - p \right | + q$
Where $a$ is the slope, $p$ is the horizontal translation, and $q$ is the vertical translation.
You can also use combinations of absolute values to include more 'sharp'/discontinuous points.
Be aware that absolute values are either defined by piecewise functions:
$\left | x \right | = \begin{cases} -x &\text{if }x<0\\ x&\text{if } x\geq 0\end{cases}$
Or:
$\left | x \right | = \sqrt{x^2}$
As far as I know, there are no exact functions which match your definition, but you can approximate them. However, I would presume that these 'approximation functions' are quite insanely complex.