December 17, 2016

This Demonstration shows a family of continuous functions that converge uniformly to the zero function on the real line but whose integrals are all equal to 1. Thus, the limit of the integrals over the real line is not the integral of the limits. However, by fixing the endpoints (the movable bluish points on the real axis) and then increasing ... , you see that the integrals over a fixed finite interval do tend to 0. Moreover, by the dominated convergence theorem, such a family of functions cannot be simultaneously bounded (over almost all the real line) by an integrable function. We illustrate this by showing two examples of functions that have a finite integral over the whole real line, which fail to bound ... for large enough ... . The two examples are ... , whose integral is equal to ... , and ... , whose integral is equal to 4. The graphs of both functions "eventually" fall below the graph of ... for large enough ... and large enough values of ... .