Approximation of continuous function by differentiable functions

Question

Let $\varepsilon > 0, t \in (0,\infty), f(x) = \begin{cases}1 \text{ if } x < t \\ 1 - (x-t)/\varepsilon \text{ if } t \le x \le t + \varepsilon \\0 \text{ else }\end{cases}$. I want to approximate this function by $C^1$ functions, so that a sequence of $C^1$ functions converges pointwise to f. There are two points where the function is not differentiable: $t, t+\varepsilon$. Each of them can be approximated separately by $C^1$ functions. For example: $1 - \frac{1}{\varepsilon} \text{sgn}(x-t)(\sqrt{(x-t)^2+\delta^2}-\delta) \to 1-(x-t)/\varepsilon$. But now I have the problem that I approximated only one of the two points. If I do the same for the other point, I don't know how to connect the two functions. Is it possible to approximate the whole function by a concrete sequence of $C^1$ functions, or do I have to work with f. e. mollifiers or something else ?