A set valued function \(U:{\mathbb{R}}\to 2^ X\) (where X is a real normed space) is said to be quadratic iff \(U(s+t)+U(s-t)=2U(s)+2U(t),\) for all s,\(t\in {\mathbb{R}}\). There is proved, among others, that if a quadratic set valued function U:\({\mathbb{R}}\to CC(X)\) (where CC(X) denotes the family of all compact, convex and non-empty subsets of X) is bounded on a subset of \({\mathbb{R}}\) of positive inner Lebesgue measure or if it is measurable, then it is of the form \(U(t)=t^ 2U(1),\) \(t\in {\mathbb{R}}\).