hi,
see here: Fixed Point Theorem -- from Wolfram MathWorld
C means that there always exists a point $\displaystyle x_{0}\in[0,1]$such that $\displaystyle f(x_{0})=x_{0}$ or in other words $\displaystyle =f(x)$ intersects $\displaystyle y=x$ at least once (try to sketch a graph).
the answer is D, let $\displaystyle f(x)\equiv1/2$.