Limit everywhere function

Here is a strange math problem.

Suppose $f:[0,1] \to [0,1]$ is function whose limit exists everywhere.

i.e $g(t) = \lim_{x \to t} f(x)$ is well defined, and gives rise to another function.

Now suppose $f$ and the limit function $g$ satisfy

$$f(x) = 2g(x) - \sin x \quad \forall x \in [0,1]$$

Find all such $f$.

[Solution]