A sequence of smooth functions such as $h_n (x) = arctan(nx)$ has discontinuous limit that depends on x. When $n \rightarrow \infty$, $h_n (x)$ converges to $$\left\{\begin{array}{ccc} -\pi/2, & \quad & x < 0\\ 0, & \quad & x = 0\\ \pi/2, & \quad & x > 0 \end{array}\right.$$
In order to show that we would evaluate three limits when $x<0, x=0$ and $x>0$.
We would do it similarly for a continuous sequence of piecewise linear functions such as
$$f_n (x)= \begin{cases} 0, x\leq0 \\[2ex] nx, 0<x<\frac{1}{n} \\[2ex] 1, x\geq \frac{1}{n} \end{cases}$$
Here we have explicitly given sequences of functions $h_n(x)$ and $f_n(x)$ and we can separate and evaluate limits on three cases based on the value of $x$.
Let's say that now we know the sequence of smooth functions that should converge to the discontinuous limit function, when some parameter(s) goes to zero. But on one hand this sequence of smooth functions is not given explicitly, i.e. it consists of some operator valued functions (e.g. some Banach space-valued functions). On the other hand the discontinuous limit function is given explicitly i.e. it is not a Banach space-valued function? In the case I am interested in, the limit function is:
$$ u(x,t)= \begin{cases} u_l, x<\mu t \\[2ex] u_r, x>\mu t \end{cases}$$
(here $u_l$, $u_r$ and $\mu$ are constants). How would we show convergence here?
Two examples above are given just for ilustration. But the PDE problem I am currently dealing with comes down to this. Help with this is appreciated.