1.7: Continuity

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Continuity

Continuity is an important concept in the theory of real functions. A continuous function is one whose output \(f(x)\) does not undergo abrupt jumps when \(x\) changes by tiny amounts. A function can be continuous over its entire domain, or only a subset of its domain. For example, \(\sin(x)\) is continuous for all \(x\), whereas \(f(x) = 1/x\) is discontinuous at \(x = 0\). Another function that is discontinuous at \(x=0\) is the step function \[\Theta(x) = \left\{\begin{array}{ll} 1, &\;\;\;\textrm{for} \; x \ge 0\\ 0,&\;\;\; \textrm{otherwise.}\end{array}\right.\] Mathematicians have even come up with functions that are discontinuous everywhere in their domain, but we won’t be dealing with such cases.

The rigorous definition of continuity is as follows:

Definition: Word

A function \(f\) is continuous at a point \(x_0\) if, for any \(\epsilon >0\), we can find a \(\delta >0\) such that setting \(x\) closer to \(x_0\) than a distance of \(\delta\) brings \(f(x)\) closer to \(f(x_0)\) than the specified distance \(\epsilon\).

That’s a very complicated sentence, and it may be easier to understand using this illustration:

A counter-example, with a function that has a discontinuity at some \(x_0\), is shown below:

If we choose \(\epsilon\) smaller than the gap, then no matter what value of \(\delta > 0\) we try, any choice of \(0 < x < \delta\) will give a value of \(f(x)\) that’s further than \(\epsilon\) from \(f(x_0)\). Hence, the continuity condition is violated for sufficiently small choices of \(\epsilon = 1/2\), and we say that \(f\) is discontinuous at \(x_0\).