How to prove continuity of $f(x)=m(A\cap (-\infty ,x])$

Question

I need to prove the continuity of the function $$f(x)=m(A\cap (-\infty,x])$$ being $m$ the Lebesgue measure function, $x\in\mathbb{R}$ and $A\subset \mathbb{R}$ a measurable set so that $m(A)=1$.

I'm sure i have to use the definition of continuous function, but i'm not sure how to do it. I will thank any help or hint.

Answer 1

If $x<y$ then $|f(y)-f(x)|=f(y)-f(x)=m(A \cap (x,y]) \leq m( (x,y])=y-x$ so $f$ is uniformly continuous.