  It is a classical fact that the exponential function is solution of the
integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this
equation to $ \int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it
seems that no classical techniques apply to yields solutions. In this article,
we consider the parameter $\alpha=1/2$. We will show the existence of a
solution wich takes the values of the Thue-Morse sequence on the odd integers.
