Limit and Continuity (20 points)

The trick is immediate value theorem.
If f(y) = k =/= 0.5, where k is rational.
Then, between 0.5 and y, f(x) must be able to attain any number between k and 0.5. But that includes the irrational number:
0.5 * 1/(1+ sqrt(2)) + k * sqrt(2)/(1+sqrt(2)).
=[0.5 * (1 - sqrt(2)) + k * sqrt(2) * (1 - sqrt(2))] / -1.
=[0.5 - 2k + (k-0.5)sqrt(2)]/(-1)
Which is evidently irrational, which lead to contradition.
Therefore f(x) = 1/2 for all x between [0, 1].