Weird function that changes digits

Every real number $a$ has one only decimal representation a = $\sum_{n \in \mathbb{Z}} a_{n} \cdot 10^{n}$ , $10 > a_{n} \ge 0 , a_{n} \in \mathbb{N} \Rightarrow$ $g:\mathbb{R} \rightarrow \mathbb{R}$ is a strictly increasing function(and $\mathbb{R}$ is an interval) $\Rightarrow$ the set of discontinutity points of $g$ is at most countable infinite. Furthemore, g is injective and g can't be continuous because then $g$ will be a bijective function and superjective function with its image(range) which would be an interval (appplying value mean Bolzano Theorem or the gontinuous image of a conected(conex) set is a conected set)but, there exists x and y such that g(x) < 4.9 < g(y) and 4.9 doesn't belong to the range of $g$ $\Rightarrow$ there must exists z such that x < z < y where g is discontinuous at z(4.9 isn' t the image for g at any point) (so 4.9, 5.29, 6.39.. don't belong to the range(image) of g) $\Rightarrow$ the points of discontinuity of $g$ is countably infinite