Value of summation of function

[man111]

If [tex]f(x)[/tex] is a differentiable function such that [tex]f:\mathbb{R}\rightarrow \mathbb{Q}\;,[/tex] Then [tex]\displaystyle \sum^{100}_{r=0}(-1)^rf(r)[/tex] is

[nevrodermit]

Since differentiability implies continuity we have that [tex]f[/tex] is continuous.
Suppose that [tex]f[/tex] is not a constant, so there exists [tex]a,b\in \mathbb{R}[/tex] such that [tex]f(a)<f(b)[/tex]. By the density of the irrational numbers we have that there are infinitely many irrational numbers in [tex][f(a),f(b)][/tex]. Let [tex]ir\in [f(a),f(b)], ir\in \mathbb{R}/\mathbb{Q}[/tex]. By the intermediate value theorem [tex]\exists c\in \mathbb{R}: f(c)=ir[/tex] which is a contradiction with the definition of [tex]f[/tex] to be to the rationals.

So [tex]f(x)=c[/tex] for some constant [tex]c\in \mathbb{Q}[/tex]. And so the given sum is [tex]\sum_{r=0}^{100}(-1)^rf(r)=[f(0)-f(1)]+[f(2)-f(3)]+\cdots + [f(98)-f(99)]+f(100)=f(100)=c[/tex].