I have the solutions book and see the answers, but they make no sense to me.

Any/All help is appreciated.

a) Find a function that is discontinuous at 1, 1/2, 1/4, 1/4... but continuous at all other points.

b) Find a function that is discontinuous at 1, 1/2, 1/4, 1/4... and at 0 but continuous at all other points.

Problem 7 doesn't seem to have enough explanation in the solution either. How is f being continuous 0 used?

7) Suppose f satisfies f(x+y) = f(x) + f(y) and f is continuous at 0. Prove it is continuous at A for all values of A.