Real examples of discontinuous functions #
A first analysis course formally develops the notion of a continuous function. Many physical quantities that we measure can be thought of as continuous functions: temperature and barometric pressure at a location are continuous functions of time; so is the distance an object has fallen after it has been dropped.
Give an example of a discontinuous function \(f:\mathbb{R}\rightarrow \mathbb{R}\) that comes up in the real world. Reconcile it with one of the characterizations of continuous functions we know from analysis, for instance, the \(\delta-\epsilon\) or the sequential definition.
Be ready to share your example with the class. You can peek below if you want some inspiration.
Solution
My favorite example #
The falling stick is my favorite example of such a function. Position a stick roughly perpendicular with the ground. If we let it go, it can
- fall ninety degrees to the left,
- fall ninety degrees to the right, or
- remain standing upright.
The last option is unlikely; it will occur if you placed the stick exactly upright. What I just described is indeed a function: the input \(\theta\) is the angle the stick makes with the ground (measured in degrees from the right); the output \(f(\theta)\) is the angle of the stick makes with the ground (again measured from the right) after you have let it go and waited a few seconds.
Image by Maia Pietraho
This function is not continuous at \(\theta=90\). To make sure this jibes with our formal understanding of continuity, we recall the sequential characterization:
A function \(f:\mathbb{R}\rightarrow \mathbb{R}\) is continuous at \(x_0\) iff whenever \(\lim x_n = x_0\) then \(\lim f(x_n) = f(x_0)\).
In plainer English, for a continuous function, we can approximate the value of \(f(x_0)\) by using \(f(x_n)\). In our stick example, if we use a sequence of angles all just larger than ninety degrees \(\theta_n\) that approach ninety, our approximation would always yield \(f(x_n) = 180\); that is, the stick falls to the left. But \(f(90) = 90\); that is, the stick actually remains upright.
So suppose that \(\mathcal{D}\) is generated by a discontinuous function, that is, \(f(x) = y\) for pairs \( (x,y) \in \mathcal{D} \). Even if we know the prediction that \(f\) should make perfectly at a large number of values of \(x\), in general one can say very little about what the value of such an \(f\) ought to be at even nearby points! If a function \(f\) is not continuous, then we may have
\[ \lim x_n = x \; \; \; \text{ but } \; \; \; \lim f(x_n) \neq f(x).\]
So even if each \(x_n\) is an input in \(\mathcal{D}\), we cannot in general hope to make an accurate prediction of the value of \(f(x)\) unless \(x\) is itself an input in \(\mathcal{D}\). Consequently, we will focus our efforts on the study of continuous functions.