Here's two problems that troubled me a lot more than they should

Problem A: Find a continuous function g:R→R and a Lebesgue measurable function h:R→R such that h∘g is not Lebesgue measurable.

Problem B: Find two a.e. continuous functions g,h:R→R such that g∘h nowhere continuous.

These would be normally easy if one writes the definitions down - but took me a lot of effort to get an answer...
Can you guys prove it in a couple of lines?

Please don't. I'll feel old.