A question on bounded variation and sum of absolute values at points of extrema.

Question

I have a function $f$ defined on $(-a,a)$ as $f(x) = 1$ when $x = 0$ and $0$ elsewhere. There is a sequence of smooth functions $\{f_n\}$ defined on $(-a,a)$, such that $f_n\to f$ pointwise. Also $V(f_n)\to V(f) = 2$ where $V(f)$ is the variation of function $f$ in the inetrval $(-a,a)$.

Lets define the set of points of extrema of function $f_n$ as $E_n = \left\{x/x \in (-a,a)\wedge f_n'(x) = 0\right\}$. and $$\beta_n = \sum\limits_{x\in E_n}\left|f_n(x)\right|$$

I am tempted to say that $$\lim_{n\to\infty}\beta_n = 1$$

Is this justified?

Answer 1

No. For example, $f_n$ could have two maxima near $x=0$ with a local minimum at $x=0$.