A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation $$frac{f(x)-F(y)}{x-y}=M(g(x),;G(y)),;x{ eq}y$$, where M is a given mean and $f$, F, $g$, G are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.