Weird Functions; why continuity matters so much

This summer I have been working through a delightful book by Gelbaum and Olmsted called Counterexamples in Analysis. No, I don't own stock in the Dover book company, nor do I know either of the authors.

I can recommend this book to anyone who is either learning elementary analysis, interested in learning the "whys" behind calculus, or is teaching calculus, real analysis or complex variables.

I'll share a few cool examples that I was not familiar with prior to reading this book.

• A function that is continuous precisely at the irrational numbers whose discontinuities are removable.

A reminder: if the limit as "x" approaches "a" of a function "f" exists but is not equal to the value of the function at "a", we say that the discontinuity is "removable". For example, the function f(x) = (x^2 -1)/(x + 1) has a removable discontinuity at x = 1 because f(1) doesn't exist (zero in the denominator) but the limit of f as x approaches 1 is clearly zero.

When we bring this concept up in calculus, we show a graph with a single "hole" where (a, f(a)) should be.

Now recall that a rational number is a number that can be put in the form p/q where p and q are integers and p, q have no common factors other than 1 or -1. If we insist that, say, q be positive, then this form is unique.

Now define f(x) to be "1/q" if x = p/q (i.e., x is rational) and let f(x) = 0 if x is irrational (e. g. f(pi) = 0).

Then note that f is continuous at every irrational number; it is a wonderful delta-epsilon argument to see this. Also note that if r is any rational number and we redefine f(r)=0 and leave f alone elsewhere, f becomes continuous at r. Hence each discontinuity is removable.

• A function that is bounded at every point but is unbounded on any interval (closed or open; we don't allow for empty or single point intervals)

Now let g(x) = q if x is rational and x = p/q and let g(x) = 0 if x is irrational.

Note that any interval whatsoever contains rational numbers of arbitrarily high magnitude (to see this, divide the real number line into, say, 1/10'ths, 1/100'ths, 1/1000's and so on, the way you would a ruler). Hence g is unbounded on any given interval, even though g(x) is finite for any given x.