The functions \(f\) and \(g\) are defined for all non-integral values and are continuous there.

Both \(f\) and \(g\) are periodic of period \(1\), that is \(f(x + 1) = f(x)\) and \(g(x + 1) = g(x)\) hold for all \(x\in \mathbb {R}\setminus \mathbb {Z}\).

Both \(f\) and \(g\) are odd functions, that is we have \(f(-x) = -f(x)\) and \(g(-x) = -g(x)\) for all \(x\in \mathbb {R}\setminus \mathbb {Z}\).

The two functions \(f\) and \(g\) sarisfy the same functional equation: \(f(\frac{x}{2}) + f(\frac{x + 1}{2}) = 2f(x)\) and \(g(\frac{x}{2}) + g(\frac{x + 1}{2}) = gf(x)\).

By setting \(h(x) \colonequals 0\) for \(x \in \mathbb {Z}\), \(h\) becomes a continuous function on all of \(\mathbb {R}\) that shares the properties given in 26.2, 26.3, 26.4.