Showing measuability of a function

Question

I want to show that the function $f:\mathbb R\to \mathbb R$ defined by $f(x)=x^2+\sin x$ for all $x\in \mathbb Q$ and $f(x)=x^2-\cos x$ for all $x\in \mathbb R\setminus \mathbb Q$ is measurable. I am not getting any way out. I can not show that this $f$ is the derivative of a function as its points of continuity is a countable set and so of first category. Please suggest!

Answer 1

The set of measurable function is an algebra ! Therefore, sum and product of measurable function are measurable. In your case, $$f(x)=(x^2+\sin(x))\cdot 1_{\mathbb Q}(x)+(x^2-\cos(x))\cdot 1_{\mathbb R\backslash \mathbb Q}(x),$$ which is sum and product of measurable function. Therefore, it's measurable !