Course:MATH110/Archive/2010-2011/003/Notes/Continuity

When computing limits in the previous sections, we've quite often encountered functions for which you can simply plug-in the value to compute the limit. Functions with this very interesting property are called continuous functions. More precisely:

Formal definition

A function $f$ is continuous at the point $x=c$ if:

$\lim _{x\rightarrow c}f(x)=f(c)$

We say that a function is continuous on the open interval ($a$, $b$) if it is continuous at any point $c$ in that interval.

Visual characteristics of continuous functions

Visually, we see that functions that are continuous on their domains can be described as the functions which can be drawn without lifting your pen.

Examples

Most functions you might write naturally are continuous (unless you break them in several distinct cases). So for examples, the following functions are all continuous on their domains:

• polynomial functions, such as:

$\displaystyle {f(x)=4x^{12}-12x^{3}+7}$,

• rational functions, such as:

$g(x)={\frac {4x^{12}-12x^{3}+7}{x^{2}-16}}$

• trigonometric functions, such as:

$\displaystyle {h(x)=\sin(x)}$,

• root functions, such as:

$\displaystyle {k(x)=x^{1/3}}$,

Operations on continuous functions

To make things even better, continuous functions are very friendly towards each other. If you start with two continuous functions, there are a lot of operations you can perform that preserves the continuity of the resulting function. For example, you can:

• add them
• substract them
• multiply them together
• divide one by the other (as long as the one you divide with never takes zero values of course)
• compose them

Conditions for continuity

For a function $f$ to be continuous at a point $x=a$, we need to satisfy the three following conditions:

• The function must be defined in some interval around the point of interest. For example, if you want to know if a function is continuous at $x=3$ it should be not only defined at that point (so, $f(3)$ should exist for sure), but actually at least a little bit around it, say for example on the interval [2,4].
• The function must admit a limit at that point, or in symbols:

$\lim _{x\rightarrow a}f(x)$

must exist. Which is equivalent to ask for both the limits on the left and the right to exist and to have the same value, or in symbols:

$\lim _{x\rightarrow a^{-}}f(x)=\lim _{x\rightarrow a^{+}}f(x)$

• Finally, we need that this limit corresponds to the actual value of the function at the point we're looking at, i.e.:

$\lim _{x\rightarrow a}f(x)=f(a)$

So there are at least three different things that can make a function not continuous at a point, you should be able to find examples for each. Use these examples to remember what it takes for a function to be continuous, that's how mathematicians do, they don't memorize things, they think of examples that work or don't. Next time you look at a function that's continuous, ask yourself if all three conditions are satisfied, how and why.

Properties of continuous functions

Continuous functions are pretty cool. They really do things that seems very intuitive. If you think of it, that idea that the limit of the function as it gets close to a point should actually be the value of the function at that point is really what you would expect. It really feels like functions which don't do that are the weird ones. This is why continuous functions are the only one that have really neat and almost intuitive properties. Let's mention a few of these properties:

The Intermediate Value Theorem

This theorem tells us that when you have a continuous function defined on an interval, say the interval [0,10] and if your function takes the value -3 at 0 (i.e. $f(0)=-3$) and the value 12 at 10 (i.e. $f(10)=12$), then the function must take at least all the values between -3 and 12. In other words, if you think of any number $u$ between -3 and 12, then there has to be at least one value $c$ between 0 and 10 for which $f(c)=u$.

We often use this theorem to guarantee that a function will intercept the $x$-axis. To see if you understand this, try to use this to construct a very solid and precise argument explaining why if I take any function $g$ which is a polynomial of degree 3 (for example, think of $g(x)=x^{3}+3x^{2}-x+4$) then there must always exist at least one solution to the equation $g(x)=0$, but it's not necessarily true if I take a polynomial of degree 4. (How can we generalize this fact?).

The Boundedness Theorem

This theorems says another fact that seems at first very obvious but which can only be true for continuous functions (try to find counter-examples!). If I look at a continuous function defined on an interval [a,b], then that function has to be bounded on that interval. That means that there must exists a maximal value $M$ and a minimal value $m$ in between of which all the values of $f(x)$ stay. We can write that in a more compacted way as follow:

$m\leq f(x)\leq M$    for all values $x$ such that    $a\leq x\leq b$

When reading a theorem like this, it is worth trying it out for yourself. Take a piece of paper and start drawing! Can you find counter-examples? (you should not, but it's worth trying). Can you create functions that are not bounded on an interval? Then according to the theorem, they shouldn't be continuous, do you see it? Do you see why? That might lead you to a hint why continuity is so important for this property to be true, maybe you can even find a proof for this theorem.

Both of these two theorems were proven for the first time by the Bohemian mathematician and priest Bernard Bolzano in 1817. Those properties were initially thought to be to too intuitively obvious to require a proof until a more precise distinction of continuous and non continuous functions was made hence offering opportunities for some counter-examples.

The Stone-Weierstrass Theorem

This theorem is already deeper but has millions of applications. It's fairly old, but it started being very useful when we invented computers who can compute easy things very quickly. What does it say? It says that when you look at a continuous function defined on an interval, then you can approximate (on that interval) as precisely as you want that function using only polynomials. If you really think of it, it's pretty neat because polynomial functions are really easy to compute (just some multiplications and additions) which what computers are good at. So whatever your function, however complicated it might be to compute its values, you can always replace it by some nifty polynomial which will be just as good (just tell us how precise you want to be, the theorem will tell you precisely how to find the adequate polynomial) and voilà! The corresponding Wikipedia article has some more details about it if you want to know more.

This theorem was initially proved by the father of modern analysis, the German mathematician Karl Weierstrass in 1885. It got then considerably generalized and its proof simplified by the American mathematician Marshall Stone betwen 1937 and 1948. Some more recent generalizations of this theorem are as recent as 1965.

Related material

Section 2.6 is the one on continuity, but again, what is written above is way enough, don't confuse yourself with the textbook, instead focus on its examples and problems. If you get stuck, write something down and try something else, you can always ask in class or on the wiki to see what's going on. Try for yourself examples 1 - 6. Just read the question and then try alone, take some notes and then only after read what the textbook has to offer. Does it match your own work? If not, why? Then, practice with some problems. Do problems 1, 3, 7, 9-12, 13-18, 19-24, 29-32, 35-40.