We study the notion of {\it finitely determined functions} defined on a topological vector space $E$ equipped with a biorthogonal system. We prove that, for real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are many finitely determined nowhere continuous functions. This notion will be used to obtain a necessary and sufficient condition for a convex function to attain a minimum at some point. An application to the Karush-Kuhn-Tucker theorem will be given.