Hey all, having trouble with part (vi) and (vii) for this. For part (v) I used f(x) = x if x is rational and -x if x is irrational (works for x0 = 0 since it is continuous at 0) and was hoping I could use that strategy for (vi) and (vii) but don't think it works, the countably infinite part is throwing me off. Could anyone help?

Thanks

(vi) Let E(x) the integer part of x.

Try with f(x) = x * E(1/x) if x is non zero, f(0) = 1.

(the x part is only here to make f continuous at x₀ = 0)

(vii) I can think of an example (there are probably easier possibilities) :

Let g a piecewise linear function defined as follows :

g(0) = 0,
g(1/n) = 0 if n is an odd integer,
g(1/n) = 1/n if n is an even integer (except 0 of course),
g is linear on every interval of the form [ 1/(n+1) ; 1/n ] if n > 0, and of the form [ 1/n ; 1/(n+1) ] if n < 0.

Now multiply g by characteristic function of Q (the function f you used for (v) ).

f = g * χQ

If should work : f is continuous at every 1/n with n an odd integer, and continuous at 0, and discontinuous everywhere else.

Thank you feanur!