Solution:

Note that $$|f(f(400))-f(f(300))|$$ $$\leq \dfrac 12 | f(400) - f(300)|$$ $$\leq \dfrac 14 |400-300| = 25,$$ and $$|f(f(900))-f(f(800))|$$ $$\leq \dfrac 12 | f(900) - f(800)|$$ $$\leq \dfrac 14 |900-800| = 25.$$

Since $f(900) = f(300),$ by the triangle inequality, we know $$|f(f(800)) - f(f(400))| =$$ $$|(f(f(800)) - f(f(900))) -$$ $$(f(f(400))-f(f(300)))| \leq$$ $$|(f(f(800)) - f(f(900)))| +$$ $$|(f(f(400))-f(f(300)))|$$$$\leq 50.$$

Now, we must conclude this value is attainable. We can make $f(x)$ a piecewise function such that $f(x) = 600$ if $x > 900$ or $x< 300,$ $$f(x) = -\dfrac 12 (x-300)+600$$ if $300 \leq x \leq 400,$ $$f(x) = \dfrac 12 (x-600)+600$$ if $400 < x < 800,$ and $$f(x) = -\dfrac 12 (x-900)+600$$ if $800 \leq x \leq 900.$ This would make $f(f(400)) = f(550) = 575$ and $f(f(800)) = f(650) = 625.$ This yields a difference of $50,$ so our result holds.

Thus, the answer is B .