Let $f$ be your two-piece linear function. Let $\varphi\in C^\infty_0((-\epsilon,\epsilon))$ for some small $\epsilon$, such that 

* $\varphi$ is even
* the integral $\int \varphi = 1$
* $x\varphi' \leq 0$ 

Then you can check that the convolution $\varphi*f$ is increasing, smooth, and agrees with $f$ outside $(-2\epsilon,2\epsilon)$. 

Taking appropriately rescaled versions of $\varphi$ you get uniform approximations.