For each of the following, define a function satisfying the
conditions, graph your function.
- $ g(x)$ is a rational function which is defined for all $x$ {except} $x=1,2$.
- $\lim_{x \to 0}{ f( x)}$ does not exist and $|{ f( x)}|<2$ for all $x$.
- $ f( x)$ is discontinuous at every point.
Tags: continuity, abstract-function, creating-examples, a--a, g classic, t1, c3
These are three unrelated pieces, really. (a) is a standard find-the-rational function problem, (b) a jump discontinuity and then there is (c). (c) has a number of answers, but the standard one is 1 at the rationals, 0 at the irrationals, of course. Most students who have had calc before (e.g. college students, post-AP) should have at least a partial notion of how to do this. But if students haven’t seen that yet, this will be a stumper. Some might even complain that it’s not a “real” function. Think about how you want to deal with that.
– Eric Hsu 1998-08-09