What about the function of real number f(x) where 0 when x is an uncomputable real number(chaitins number) and 1 if its computable like π.
Its certainly not continuous anywhere What else can we know about this function?
In the spirit of Dirichlet function which can be construct using limits of sequence of functions, is there a way to approximate this?
In a precise sense, your function $f$ is no more complicated than the Dirichlet function $D$.
Cantor showed that any two countable dense linear orders without endpoints are isomorphic. Let $i$ be an isomorphism between the rationals and the computable reals; by density, $i$ extends uniquely to a homeomorphism (= continuous bijection with continuous inverse) $\hat{i}:\mathbb{R}\rightarrow\mathbb{R}$. This map has the property that it "swaps" $f$ and $D$: $$D=f\circ \hat{i}\quad\mbox{and}\quad f=D\circ \hat{i}^{-1}.$$ So $f$ and $D$ are "topologically the same."