ABSTRACT
This chapter deals with continuous functions, indeed abstracted from our experience in the set of real numbers. This chapter starts with the usual definition of continuity, and then we form some equivalent definitions which help us proving or disproving the continuity of any function. This is the point, where the keen reader is ready to start learning general topology.
After looking at sufficient examples and exercises of continuous functions, we “upgrade” this idea to uniform continuity and Lipschitz continuity (although, we only scratch the surface of Lipschitz continuity without going into depth). We then look at the consequences of applying continuous functions on various types of metric spaces (compact, connected, complete, bounded, etc.) dealt with earlier in the text. Finally, as a close of this chapter, the reader is introduced to Homeomorphisms, the very core of general topology and all the theory that develops further.
