International Mathematical Forum, Vol. 8, 2013, no. 16, 783 - 788 HIKARI Ltd, www.m-hikari.com Continuity of Darboux Functions Nikita Shekutkovski Ss. Cyril and Methodius University, Skopje , Republic of Macedonia nikita@pmf.ukim.mk Beti Andonovic Ss. Cyril and Methodius University, Skopje , Republic of Macedonia beti@tmf.ukim.edu.mk Copyright c  2013 Nikita Shekutkovski and Beti Andonovic. This is an open access article distributed under the Creative Commons Attribution License, which permits unre- stricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In 1965, Whyburn proved that if X is locally connected and first countable, Y is Hausdorff, and f: X→ Y a function, then: f is continous iff f preserves compactness and connectedness. Definition: A function f: X→ Y is preserving path connectedness (a Darboux function) if the image of any path-connected subset of X is path connected. As an example, the derivative f ’ of a real differentiable function f defined on an interval is path preserving although f ’ is not always continuous. In the paper we prove the following theorem Theorem: Suppose X is Hausdorff,locally path-connected and 1- countable, Y is Hausdorff, and f: X→ Y a function. Then f is continuous iff f preserves compactness and f preserves path connectedness. By elementary theorems if f : X → Y is a continuous function then the image of compact subspace of X is compact set, and the image of connected set is connected. In 1970 McMillan proved that if X s Hausdorff, locally con- nected and Frechet, Y is Hausdorff, then the converse is also true. In fact McMillan generalized Whyburn’s theorem from 1965. The last advance in this direction is made in [2] by the following result of Whyburn [5]: Suppose X is locally connected and 1-countable, Y is Hausdorff, and f : X → Y a