I want some help in building a strategy to prove/disprove a statement in a specific problem. The specific problem setup is invisible to this question.
In my problem, I have a sequence of smooth, analytic functions $\{f_n\}$, which converges pointwise to the function $g$ which is like $g(x) = 0,x\notin \{x_1,x_2,x_3\}$ and $g(x_1) = a_1,g(x_2) = a_2, g(x_3) = a_3$. I want to prove/disprove that $$\lim_{n\to\infty}TV(f_n) = \left|a_1\right| + \left|a_2\right| + \left|a_3\right|$$ by leveraging on certain properties of this function sequence. Here $TV(f)$ is the total variation of $f$.
I know the above statement is not true in general, but there are many properties on this function sequence, in the main problem set up, and I want to leverage them and see if the above statement I made, holds or not. I want a general strategy that I can employ in this regard. I know it may not make complete sense as the main problem is invisible, but I am stuck and want to see if there is a general strategy for this kind of setup. Appreciate some help in this regard.