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Continuous functions from a subset of $\mathbb{R}$ to $\mathbb{R}$

Graphs can fool you

Continuous functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}$

Continuous functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$

Continuous functions from a subset of $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}$

Continuous functions from a subset of $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$



Functions can be represented in many different ways. There is a sharp difference between representing continuous functions and representing discrete functions.

As you will see in Representations of Discrete Functions, many different arrangements of the inputs and outputs for discrete functions can be made, but none of them can use the idea that close inputs give close outputs. On the other hand, a representation of a finite function can be absolutely accurate.

The lllustrations in this post

The illustrations were created using these Mathematica Notebooks:

Continuous functions from a subset of $\mathbb{R}$ to $\mathbb{R}$

Graphs of functions of one variable

The most familiar representations of continuous functions are graphs of functions with one real variable. Students usually first see these in secondary school. Such representations are part of the subject called Analytic Geometry. This section gives examples of such functions.


The blue curve in the diagram below is a representation of the graph of the function $g(x):=2-x^2$ for approximately the interval $(-2,2)$.

The brown right-angled line in the upper left side, for example, shows how the value of independent variable $x$ at $(0.5)$ is plotted on the horizontal axis, and the value of $g(0.5)$, which is $1.75$, is plotted on the vertical axis. So the blue graph contains the point $(0.5,g(0.5))=(0.5,1.75)$.

Fine points

Discontinuous functions

A function which is continuous except for a small finite number of breaks can also be represented with a graph. Such functions are called piecewise continuous functions.


Below is a graph of (part of) the function $f$ defined by\[f(x):=\left\{ \begin{align} 2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\gt0) \\ 1-x^2\,\,\,\,\,\,(-1\lt x\lt 0) \\ 2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\lt-1) \end{align}\right.\] $f$ is one function. It is defined for every real number except $-1$ and $0$.

Graph of a discontinuous function.

Another discontinuous function is shown in Examples of functions


This example uses the concepts of rational and irrational.

The Dirichlet function is defined by\[F(x):= \begin{cases} 1 & \text{if }x\text{ is rational}\\ \frac{1}{2} & \text{if }x\text{ is irrational}\\ \end{cases}\]for all real $x$.

It is impossible to draw a graph of this function, because it has an infinite number of gaps between its points. For more detail, see the article Examples of functions.

Graphs can fool you

The graph of a continuous function cannot usually show the whole graph, unless it is defined only on a finite interval. It is also only approximately accurate: You cannot determine precisely the exact location of any point on the graph. These faults can lead you to jump to incorrect conclusions.


For example, you can't tell from the the graph of the function $y=2-x^2$ whether it has a local minimum (because the graph does not show all of the function), although you can tell by using calculus on the formula that it does not have one. The graph also looks like it might have vertical asymptotes, but it doesn't, again as you can tell from the formula (it is defined for all real numbers).

Discovering facts about a function
by looking at its graph
is useful but dangerous.


Below is the graph of the function\[f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\]

If you didn't know the formula for the function, by looking at the graph you might nevertheless suspect certain things are true of the function.

  1. Maybe it has a local maximum somewhere to the right of $x=1$.
  2. Maybe it has one or more zeroes around $x=-1$.
  3. Maybe it has one or more zeroes around $x=2$.
  4. Maybe it has a vertical asymptote somewhere to the right of $x=2.5$.
  5. Maybe it has a horizontal asymptote in the negative $x$ direction.

If you do know the formula, you can find out many things about the function that you can't depend on the graph to see.

The section on Zooming and Chunking gives other details.

Continuous functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}$

Suppose $F:\mathbb{R}\to\mathbb{R}\times\mathbb{R}$. That means you put in one number and get out a pair of numbers.

The unit circle

An example is the unit circle, which is the graph of the continuous function $t\mapsto(\cos t,\sin t)$. That has this parametric plot:

Unit circle

Because $\cos^2 t+\sin^2 t=1$, every real number $t$ produces a point on the unit circle. Four point are shown. For example,\[(\cos\pi,\,\sin\pi)=(-1,0)\] and \[(\cos(5\pi/3),\,\sin(5\pi/3))=(\frac{1}{2},\frac{-\sqrt3}{2})\approx(.5,-.866)\]

The input is not graphed

In graphing functions $f:\mathbb{R}\to\mathbb{R}$, the plot is in two dimensions and consists of the points $(x,f(x))$: the input and the output. The parametric plot shown above for $t\mapsto(\cos^2 t+\sin^2 t)$ shows only the output points $(\cos t,\sin t)$; $t$ is not plotted on the graph at all. So the graph is in the plane instead of in three-dimensional space.

Visualizations of the parametric plot

It is useful to think of $t$ as time. If you start at some number $t$ on the real line and continually increase it, the value $f(t)$ moves around the circle counterclockwise, repeating every $2\pi$ times. If you decrease $t$, the value moves clockwise. Every point is traversed an infinite number of times as $t$ runs through all the real numbers.

Figure-8 graph

This is the parametric graph of the function $t\mapsto(\cos t,\sin 2t)$.

Figure 8

Notice that it crosses itself at the origin when $t$ is any odd multiple of $\frac{\pi}{2}$.

Continuous functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$

The graph of a function from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ can also be drawn as a parametric graph in three-dimensional space, giving a three-dimensional curve.

The seven-pointed crown

Here is a view of the graph of the function \[t\mapsto(\cos t, \sin t, \sin 7t):\mathbb{R}\to \mathbb{R}\times\mathbb{R}\times\mathbb{R}\]

The animated gif crownmovie.gif represents the parameter $t$ in time.

Universal covers

The unit circle with $t$ made explicit

Since we have access to three dimensions, we can display on the graph the input $t$ to the unit circle function $(\cos t,\sin t)$ by using a three-dimensional graph, shown below. The blue circle is the function $t\mapsto(\cos t,\sin t,0)$ and the gold helix is the function $t\mapsto(\cos t,\sin t,.2t)$.

Unit circle

The introduction of $t$ as the value in the vertical direction changes the circle into a helix. The animated .gif covermovie.gif shows both the travel of a point on the circle and the corresponding point on the helix.

As $t$ changes, the circle is drawn over and over with a period of $2\pi$. Every point on the circle is traversed an infinite number of times as $t$ runs through all the real numbers. But each point on the helix is traversed exactly once. For a given value of $t$, the point on the helix is always directly above or below the point on the circle.

The helix is called the universal covering space of the circle, and the set of points on the helix over (and under) a particular point $p$ on the circle is called the fiber over $p$. The universal cover of a space is a big deal in topology.

Loop and swirl

Here are two views of the function \[t\mapsto(\cos t,\sin 2t, \log t):\mathbb{R^{+}}\to \mathbb{R}\times\mathbb{R}\times\mathbb{R}\] ($\mathbb{R}^{+}$ is the set of positive real numbers)

It is a universal cover (with an added squish as $z$ increases) of the Figure-8 curve.

Continuous functions from a subset of $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}$

The graph of a map $\mathbb{R}^m\to\mathbb{R}^n$ where $m+n\gt3$ requires $m+n$ dimensions, so we can't draw it and it is difficult to visualize it.

You can graph a function $\mathbb{R}\times\mathbb{R}\to\mathbb{R}\times\mathbb{R}$ using its endograph.


The function \[(x,y)\mapsto(x+.5,y+.2)\] moves each point in the plane $.5$ units to the right and $.2$ units up. The endograph below visualizes a part of this action. showing the movement at each point with integer coordinates.

This is an example of an affine map in the plane.


The linear map determined by the matrix \[\pmatrix{1.25 & 0.5\\0.5&1}\] defines a function \[(x,y)\mapsto (1.25x+.5y,\,.5x+y):\mathbb{R}\times \mathbb{R}\to \mathbb{R}\times \mathbb{R}\] that can be displayed with an endograph like this:

Complex valued functions

Complex functions are often graphed using $\mathbb{R}\times\mathbb{R}$ for the domain (representing $a+bi$ as $(a,b)$), with the values of the function shown as hue and brightness, as seen in the Wikipedia article.

Complex analysis, the theory of functions on the complex numbers, is one of the richest and most highly developed parts of math.

Continuous functions from a subset of $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$

2D surface in 3-space

This is the graph of part of the surface\[(x,y)\mapsto(x,z,\sin xz):\mathbb{R}\times\mathbb{R}\to\mathbb{R}\times\mathbb{R}\times\mathbb{R}\]


The blue curves follow the shape of the surface as an aid to visualizing the surface.

This graph was created by the Mathematica notebook AnotherFamiliesFrozen.nb It is worthwhile viewing this notebook in CDF Player, if you have it. (It is free). There are more examples in Freezing a family of functions.

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