I have a specific function in mind, but I would like to know about the general theory.
I have a sequence of functions $\lbrace W(x) \rbrace _{m\in\mathbb{N}}$. I know the following about $W$: at $W(0)=0, W(y^*) = 1/2, W(\infty) = 1$ (excuse the notation) where $y^*>0$ is some fixed constant. Further, $W$ is continuous, positive, increasing, differentiable, and in general is nicely behaved.
I know the following about the sequence of derivatives $\lbrace dW/dx \rbrace _{m\in\mathbb{N}}$: there is pointwise convergence almost everywhere to the zero function (exponentially, if that matters). The point $x=y^*$ where there isn't pointwise convergence, the derivative grows without bound. Heuristically, it seems that this means the original function $W$ is converging to the step function which is 0 from $0\le x<y^*$, 1/2 at $x=y^*$ and 1 for $x>y^*$.
I am wondering how to formalize this, and in general what one must know so as to make conclusions of this form (lifting pointwise convergence of sequence of derivatives to pointwise convergence of sequence of original function).
Thanks!
Attempted Proof (please check): Take some $r$ with $0<r<y^*$. Then on $[0,r]$ I can show the $\partial W/\partial x$ is increasing (for large enough $m$). Then on $[0,r]$ the sequence of derivatives is uniformly convergent to the zero function (since the $m$ required to make all $\partial W/\partial x$ close to zero on $x\in[0,r]$ is just the $m$ required to make $(\partial W/\partial x) (r)$ close to zero, which is fine since we have pointwise convergence). Then, on this interval $[0,r]$ we have
$$W_k(r)-W_k(0) = \int_{[0,r]} \left\lbrace \frac{\partial W}{\partial x}\right\rbrace_k \rightarrow \int_{[0,r]}\left\lbrace \frac{\partial W}{\partial x}\right\rbrace_\infty =0$$ (where the limit is as $m\rightarrow \infty$) and then using the positivity, continuity and fact that $W$ is increasing we conclude $W_k(r) \rightarrow 0$ as $k\rightarrow \infty$. This is true for any for any $0<r<y^*$. Similarly for any $r>y^*$ we run the proof on $[r,\infty)$ get uniform convergence and conclude similarly. This demonstrates almost everywhere convergence of $W$.