6.5. Uniformly Continuous FunctionsIn this part we introduce an alternative notation of continuity, the so called $\epsilon /\delta$notation. It no longer needs sequences to be involved which in fact offers the option to introduce other concepts of continuity.
Consider:
Functions are often continuous at many points of their domain. It should be emphasized that the interaction guaranteed by the $\epsilon /\delta$notation between a given $\epsilon$ and a suitable $\delta$ normally depends on the focussed point a and it is very unlikely that $\delta$ could be set uniformly for all a. The following notion thus establishes another, narrower concept of continuity.
Consider: This new notion of continuity is a special version of the old one, because:
In [6.5.4] we used the reciprocal function to show that [6.5.3] is not reversible. Surprisingly there are no counter examples among the functions on a closed interval. The following theorem is a valuable resource when it comes to estimates.
As seen in [6.5.4] the interval needs to be closed for this theorem to be valid. The next part will show that continuous functions on closed intervals indeed excel in very special properties. We end this part with a special variant of uniform continuity, the so called lipschitzcontinuity:
Consider:
Of special interest are lipschitzcontinuous functions with a lipschitz continuity constant c below 1. Because in that case the distance of two values of f is less than the distance of their invers images and with functions that map closed intervals onto themselves this will lead to a fixed point.
Consider:
