I have stuck at a problem. " does there exist any continuous function that takes every real value exactly twice?"
Intuitively, I think such function cannot exist, as $x^2$, $|x|$, although they take every possitive value exactly twice, but they take $0$ only once. I tried to apply LMVT, but cannot make a concrete logic. Please help
You're right that no such function exists. If $f$ were one then there would be two values $a < b$ such that $f(a)=f(b)=1$. Then on the closed interval $[a,b]$ the function $f$ would have a maximum value $M$. We can assume $M > 1$ (if not, then use the minimum value).
If the maximum value $M$ appeared twice, at $c < d$ then find an $x$ near $c$ such that $1 < f(x) < M$. (You can do that using continuity). Then by the intermediate value theorem, $f$ takes on the value $f(x)$ between $a$ and $c$ and between $c$ and $b$. That's three times.