continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.
I can understand what is a continuous function. A function which just don't have any holes or jumps or vertical asymptotes. But how can we prove that a small change of input in continous function causes only small change in the outputs.
What do we define as small? Why cant a function do a big jump at point X and come back gradually around X+4 etc..?