The horizontal chord of length one half

Idea. The statement says that the graph has a horizontal chord of length $\frac12$. It is not obvious at all, because we do not know where this chord sits. The trick is to introduce the function that measures the height difference of a chord.

Auxiliary function. For $x\in\left[0,\frac12\right]$, let $g(x)=f\!\left(x+\frac12\right)-f(x)$. It is continuous. We compute
$$g(0)=f\!\left(\tfrac12\right)-f(0),\qquad g\!\left(\tfrac12\right)=f(1)-f\!\left(\tfrac12\right)=f(0)-f\!\left(\tfrac12\right)=-g(0).$$

Conclusion. The numbers $g(0)$ and $g\!\left(\frac12\right)$ are opposite, so they are either both zero or of opposite signs. By the intermediate value theorem $g$ vanishes somewhere in $\left[0,\frac12\right]$, which is the claim.

Real-life version. If the temperature at midnight is the same on two consecutive days, then at some moment the temperature equals the temperature twelve hours later.

Remark. The same proof gives chords of length $\frac1n$ for every integer $n$, but not of every length: for length $\frac25$ the statement is false in general.