As an example, the functional equation $f(x+y)=f(x)f(y)$, by declaring that $f$ is continuous and differentiable, we can arrive at the unique solution $f(x)=a^x$, by first showing that $f'(x)=f(0)f(x)$ and then using basic ODE theory to declare such an $f$ exists.
I'm interested in an example of a functional equation whose only solution is a continuous nowhere differentiable function. While some gymnastics with trigonometry might arrive at a functional equation for something like the Weierstrass functions, uniqueness might be hard to prove. Are there any good "easy" examples of such functional equations? I'd strongly prefer something that uses basic operations and classical functions and not say, derivatives in the sense of formal power series.