Uniform limit of continuous functions bounded variation

Question

Prove or disprove that if $f:[a,b]\rightarrow\mathbb{R}$ is the uniform limit of a sequence of continuous functions each of which is of bounded variation, then $f$ is of bounded variation on $[a,b].$

Answer 1

The result is false as stated: Take any continuous function not of bounded variation and approximate via polynomials (Weierstrass' theorem).

On the other hand if, for example, we had $V_a^b(f_n) \leq M$ for some constant $M>0$ and all $n$ then the result is true because $V_a^b(f) \leq \liminf_n V_a^b(f_n)$. To see this just pick a partition $(x_i)$ of $[a,b]$ and compute $$ \sum_i |f_n(x_i)-f_n(x_{i-1})| \leq V_a^b(f_n). $$ Now just take $\liminf_n$ on both sides and then the supremum over all partitions to conclude. Notice that this only needs a pointwise convergent sequence $f_n$.