This problem can be done without any derivatives, actually. First observe that since (x-y)^2=(y-x)^2, we have |F(x)-F(y)|<=(x-y)^2. Given x and y with y>x, divide up the interval between them into x_0=x, x_1, x_2, ..., x_n=y evenly spaced. Then applying the observation above and the triangle inequality, since F(y)-F(x) is the sum of the F(x_(i+1))-F(x_i), we have |F(y)-F(x)|<=n*((x-y)/n)^2=(x-y)^2/n. Since n can be arbitrarily large, |F(y)-F(x)| is smaller than any positive number and hence 0, so F(y)=F(x).
Note that 2 here can be replaced with any number greater than 1; this is actually a well-known fact, that any Hölder-continuous function on the reals with exponent greater than 1 is constant. But I suppose it would not be well-known to high-school students! To be honest, I mostly only know it because of the old legend about the student who... well, here's a link: http://mathoverflow.net/questions/53122/mathematical-urban-l...
Note that 2 here can be replaced with any number greater than 1; this is actually a well-known fact, that any Hölder-continuous function on the reals with exponent greater than 1 is constant. But I suppose it would not be well-known to high-school students! To be honest, I mostly only know it because of the old legend about the student who... well, here's a link: http://mathoverflow.net/questions/53122/mathematical-urban-l...