\(\epsilon\)-\(\delta\) Definition

The \(\epsilon\)-\(\delta\) definition formalizes what it means for a function to be continuous at a point. It states that for any small margin of error \(\epsilon\) you choose, there exists a corresponding distance \(\delta\) so that whenever the input is within \(\delta\) of that point, the output of the function is within \(\epsilon\) of the function's value at that point. In essence, it guarantees that by making the input's change sufficiently tiny (\(\delta\)), the change in output will also be tiny (\(\epsilon\)), ensuring the function behaves predictably and smoothly at that point.