Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a function of bounded variation. Then it is well known that it has right and left limits everywhere and only jump discontinuities. Set $\tilde{f}(x)=f(x+)$, the right limits of $f$. Suppose $\lambda\in\mathbb{R}$ is such that $\tilde{f}(\lambda+)=\tilde{f}(\lambda-)=\tilde{f}(\lambda)$ (i.e. a point of continuity for $\tilde{f}$). Is it also true that $\lambda$ is a point of continuity of $f$?
$EDIT:$ This is wrong. In fact, consider the function $\chi_{(-\infty,0) \cup (0,\infty)}$. It has bounded variation and its "right continuous" version is just $1$ everywhere, which is continuous at $0$ while the original function isn't. The problem lies in that I assumed somehow that functions of bounded variation only have jump discontinuities, which is wrong! They might have so-called removable discontinuities, where the limits from the left and right at a point agree but they do not agree with the value of the function at that point.
Best Answer
Addendum
There is a potential flaw in the original proof (below) which assumes that convergence to right-hand limits is uniform. Here is a different proof.
Any function of bounded variation has at most countably many jump discontinuities. There are two possibilities.
(1) There are finitely many jump discontinuities in some open neighborhood of $\lambda$. In this case, there is an interval $(\lambda - \delta, \lambda + \delta)$ where $f$ is continuous except possibly at $\lambda$. For $x \neq \lambda$ we have $f_R(x) = f(x+) = f_L(x) = f(x-) = f(x).$ By hypothesis $f_R$ is continuous at $\lambda$ and, therefore, $f_R(\lambda-) = f_R(\lambda+)$.
Thus $$f(\lambda-) = \lim_{x \to \lambda -}f(x) = \lim_{x \to \lambda -}f_R(x) = f_R(\lambda-) = f_R(\lambda+) = \lim_{x \to \lambda +}f_R(x) = \lim_{x \to \lambda +}f(x) = f(\lambda+),$$
and $f$ is continuous at $\lambda.$
(2) Jump discontinuities accumulate at $\lambda$. Let $(y_n)$ and $(z_n)$ be any increasing and decreasing sequences of points, respectively, both converging to $\lambda$. Since, the sums of the jumps must be bounded by the total variation of $f$, we have
$$\sum_{k=1}^\infty |f_R(y_k) - f_L(y_k)| < \infty, \\ \sum_{k=1}^\infty |f_R(z_k) - f_L(z_k)| < \infty,$$
and $\lim_{k \to \infty} |f_R(y_k) - f_L(y_k)| = \lim_{k \to \infty} |f_R(z_k) - f_L(z_k)| = 0$.
Thus,
$$f_L(\lambda-) = \lim_{k \to \infty} f_L(y_k) = \lim_{k \to \infty} f_R(y_k) - \lim_{k \to \infty} (f_R(y_k) - f_L(y_k)) = f_R(\lambda-), \\ f_L(\lambda+) = \lim_{k \to \infty} f_L(z_k) = \lim_{k \to \infty} f_R(z_k) - \lim_{k \to \infty} (f_R(z_k) - f_L(z_k)) = f_R(\lambda+), $$
and
$$f_L(\lambda-) = f_L(\lambda+) = f_R(\lambda+) = f_R(\lambda-).$$
Original Proof
By hypothesis, $\tilde{f}$ is continuous at $\lambda$. We must have $f(\lambda-) = f(\lambda+) =\tilde{f}(\lambda)$ when $ f$ is continuous at $\lambda$.
Suppose $f$ is not continuous at $\lambda$. This implies $f(\lambda-) \neq f(\lambda+) = \tilde{f} (\lambda).$ Since the left-hand limit exists there exists a sequence $(x_n)$ converging to $\lambda$ from the left such that $f(x_n)$ does not converge to $\tilde{f}(\lambda)$.
Hence, there exists $\epsilon_0 > 0$ such that for all sufficiently large $n$,
$$|f(x_n) - \tilde{f}(\lambda)| > \epsilon_0.$$
Now choose a sequence of points $(y_n)$ such that $x_n - 1/2^{n} < y_n < x_n.$ Then $y_n \to \lambda$ and $|f(x_n) - \tilde{f}(y_n)| < \epsilon_0/2$ for sufficiently large $n$, since the right-hand limit exists everywhere (and convergence is uniform in a compact neighborhood of $\lambda$).
Thus,
$$|\tilde{f}(y_n) - \tilde{f}(\lambda)| > |f(x_n) - \tilde{f}(\lambda)| - |f(x_n) - \tilde{f}(y_n)| > \epsilon_0/2, $$
which contradicts the continuity of $\tilde{f}$ at $\lambda.$