This paper develops the very basic notions of analysis in a weak second-order theory of
arithmetic BTFA whose provably total functions are the polynomial time computable functions. We
formalize within BTFA the real number system and the notion of a continuous real function of a real
variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the
system of real numbers is a real closed ordered field. In the last section of the paper, we show how to
interpret the theory BTFA in Robinson’s theory of arithmetic Q. This fact entails that the elementary
theory of the real closed ordered fields is interpretable in Q.