# Can one recognize a function from its graph?

### Jürgen Appell

Universität Würzburg, Germany### Agnieszka Chlebowicz

Rzeszów University of Technology, Poland### Simon Reinwand

TNG Technology Consulting GmbH, Unterföhring, Germany### Beata Rzepka

Rzeszów University of Technology, Poland

## Abstract

We analyse the “interplay” between analytical properties of a real function on a metric space, on the one hand, and topological properties of its graph, on the other. In particular, we study functions with closed, compact, connected, pathwise connected, or locally connected graphs, and we give nine conditions on the graph which are equivalent to the continuity of a function. A main emphasis is put on examples and counterexamples which illustrate how significant our hypotheses are, and how far sufficient conditions are from being necessary.

## Cite this article

Jürgen Appell, Agnieszka Chlebowicz, Simon Reinwand, Beata Rzepka, Can one recognize a function from its graph?. Z. Anal. Anwend. 42 (2023), no. 1/2, pp. 203–233

DOI 10.4171/ZAA/1730