The proof of the intermediate value theorem for power series on a LeviCivita field will be presented. After reviewing convergence criteria for power series [19], we review their analytical properties [18]. Then we state and prove the intermediate value theorem for a large class of functions that are given locally by power series and contain all the continuations of real power series: using iter...

For simplicity, we adopt the following convention: a, b, c, d, r1, r2, r3, r, r4, s1, s2 are real numbers, p, q are points of E 2 T , P is a subset of the carrier of E2 T , and X, Y , Z are non empty topological spaces. Next we state a number of propositions: (1) For all a, b, c holds c ∈ [a, b] iff a ¬ c and c ¬ b. (2) Let f be a continuous mapping from X into Y and g be a continuous mapping f...

If the function f : I → R is differentiable on the interval I ⊆ R , then for each x,a ∈ I, according to the mean value theorem, there exists a number c(x) belonging to the open interval determined by x and a , and there exists a real number θ (x) ∈]0,1[ such that f (x)− f (a) = (x−a) f (1) (c(x)) and f (x)− f (a) = (x−a) f (1) (a+(x−a)θ (x)) . In this paper we shall study the differentiability ...

This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without using countable choice. The theorem is that if a pointwise continuous function has both a negative and a positive value, then it has values arbitrarily close...

The familiar Intermediate Value Theorem of elementary calculus says that if a real valued function f is continuous on the interval [a, b] ⊆ R then it takes each value between f(a) and f(b). As our next result shows, the critical fact is that the domain of f , the interval [a, b], is a connected space, for the theorem generalizes to real-valued functions on any connected space. The Intermediate ...