Problem
ALG-B3-M05-P020 Quadraticity from a Parallelogram
Let \(f:\mathbb R\to\mathbb R\) be continuous, \(f(0)=0\), \(f(1)=1\), and \(f(x+y)+f(x-y)=2f(x)+2f(y)\) for all \(x,y\). Prove that \(f(x)=x^2\).
First obtain values on integers and rationals, then use continuity.
Putting \(x=y\), we get \(f(2x)+f(0)=4f(x)\), so \(f(2x)=4f(x)\). A standard induction using the quadratic Cauchy formula gives \(f(nx)=n^2f(x)\) for integers \(n\). With \(x=1\), this gives \(f(n)=n^2\). Then from \(f(n\cdot \frac{m}{n})=n^2 f(\frac{m}{n})\), we obtain \(f(\frac{m}{n})=\frac{m^2}{n^2}\). Hence \(f(q)=q^2\) for all rational \(q\).
For an arbitrary real \(x\), choose rationals \(q_k\to x\). By continuity, \(f(x)=\lim f(q_k)=\lim q_k^2=x^2\). Therefore \(f(x)=x^2\) for all \(x\).
Final problem of the module: proves the quadratic form rather than only quoting it.