Here we will introduce some basic properties of functions from a theoretical point of view. We start with boundedness, then pass to intercepts, symmetry, and periodicity. For practical information see Boundedness, Symmetry, and Periodicity in Methods Survey.

Definition.

We say that a real functionfisbounded from belowif there is a numberksuch that for allxfrom the domainone has D(f)f(x) ≥k.

We say that a real functionfisbounded from aboveif there is a numberKsuch that for allxfrom the domainone has D(f)f(x) ≤K.

We say that a real functionfisboundedif it is bounded both from above and below.

Equivalently, a function *f* is bounded if there is a number *h*
such that for all *x* from the domain
*D*( *f* )*h* ≤ *f* (*x*) ≤ *h*,*f* (*x*)| ≤ *h*.

Being bounded from above means that there is a horizontal line such that the
graph of the function lies below this line. Bounded from below means that the
graph lies above some horizontal line. Being bounded means that one can
enclose the whole graph between two horizontal lines. The inequalities in the
definition are often shortened like this:
*f* ≥ *k*,*f* ≤ *K*,*f* | ≤ *h*

The constants *k*, resp. *K* are called the **lower**, resp.
**upper bound** for *f*. If there is one bound, then there is
infinitely many. For instance, if *f* is at most 1, then also
*f* ≤ 1.5,*f* ≤ 13,*f* ≤ 175.37,

In the following picture, the function on the left seems to be bounded, while the one on the right seems to be bounded from below but not from above, therefore also not bounded. We marked some bounds that seem to work, we also tried some upper bounds in the example on the right to suggest that no bound would work (at least assuming that the function keeps going in the suggested way).

A beginner may be excused for thinking that if a function has a bounded
domain (for instance if it is defined only on a closed interval, say,

The only case when the domain determines boundedness is a rather extreme one: If a function is defined only on a finite set, then it is automatically bounded, since it has only finitely many values - every finite set has a maximum and minimum. However, although such functions exist, they are freaks kept is the zoo of mathematical curiosities in order to trap unsuspecting students, one rarely meets them in applications.

For a subset *M* of the domain of a function *f* we can define
**boundedness on a set**. The definition is similar, we want a bound
(upper, lower, both) that is universal for *f* (*x*)*x* from *M*.

Boundednes on a set is often used and the following simple observation can come handy.

Fact.

Lefbe a function bounded on some setM. Thenfis also bounded on all subsets ofM.

What can be said about **boundedness and operations**? If we add, subtract
or multiply two bounded functions, then the outcome is a bounded function. If
we compose two functions, and the outer one is bounded, then the whole
composition is bounded.

What does not work? If we compose two functions and the inner one is bounded,
the whole composition need not be bounded. For instance, the function
*f* (*x*) = *x*^{2}*x*,*x*^{2},

Similarly, if we divide two bounded functions, the outcome need not be
bounded if the one in the denominator gets arbitrarily close to 0. Example:
*x*

When one starts considering unbounded functions, there are lots of situations, too many to list them. There are some positive statements for boundednes above/below, most of them are common sense. For instance, adding or multiplying two functions bounded from below gives a function bounded from below. If we subtract a function bounded from below from a function bounded from above, we get a function bounded from above (write the relevant inequalities, you will see it). If we do not have even this partial boundedness, then anything is possible, for instance the sum of two such functions can be bounded, bounded above, bounded below or entirely unbounded.

One way to look closer at boundedness on a set is to ask the following
question: How high and how low does *f* go on a given set *M* (a
subset of the domain)?

Definition.

LetMbe a non-empty subset of the domain of a functionf.

Iffis bounded from above onM, we define thesupremumoffoverM, denotedsup as the least upper bound of_{M}(f),fonM.

Iffis not bounded from above onM, we definesup _{M}(f) = ∞.

Iffis bounded from below onM, we define theinfimumoffoverM, denotedinf as the greatest lower bound of_{M}(f),fonM.

Iffis not bounded from below onM, we defineinf _{M}(f) = −∞.A number

mis said to be themaximumoffoverM, denotedmax if it satisfies the conditions_{M}(f),

(1) there iscfromMsuch thatand f(c) =m,

(2)for all f(x) ≤mxfromM.

If such a maximum exists, we say thatfattains its maximumoverM(atc).A number

mis said to be theminimumoffoverM, denotedmin if it satisfies the conditions_{M}(f),

(1) there iscfromMsuch thatand f(c) =m,

(2)for all f(x) ≥mxfromM.

If such a maximum exists, we say thatfattains its minimumoverM(atc).

There is an alternative way to look at these definitions. Given a function
*f* and a set *M* as above, we can consider the set of all values
of *f* on *M*, that is,

*N* = *f* (*M*) = { *f* (*x*); *x*∈*M*}.

Then
_{M}( *f* ) = sup(*N*),_{M}( *f* ) = inf(*N*),_{M}( *f* ) = max(*N*),_{M}( *f* ) = min(*N*).

From this it follows that the supremum and infimum always exist, but the
maximum and minimum need not exist; however, if they do, they agree with
supremum and infimum, respectively. We also get a direct definition of
suprema and infima. For instance,
_{M}( *f* )*s* satisfying the following:

(1) *f* (*x*) ≤ *s**x* from *M*.

(2) For every *ε* > 0*x* from *M* such that
*f* (*x*) > *s* − *ε*.

Similar definition works for the infimum. Look at these examples:

The first example:
_{M}( *f* ) = ∞,_{M}( *f* )_{M}( *f* ) = min_{M}( *f* ) = 2.

The second example:
_{M}( *f* ) = 4,_{M}( *f* )_{M}( *f* ) = 2,_{M}( *f* )

The third example:
_{M}( *f* ) = ∞,_{M}( *f* )_{M}( *f* ) = min_{M}( *f* ) = 2.

The fourth example:
_{M}( *f* ) = max_{M}( *f* ) = 4,_{M}( *f* ) = 2,_{M}( *f* )

These pictures should make clear the meaning of these four notions, they should also show that minimum and maximum need not be attained even for bounded functions on bounded and closed intervals. For more information, see the next section on continuity.

Intercepts are points where the graph of *f* crosses the axes of the
coordinate system. There are two kinds:

- *x*-intercepts are intersections of the graph and the *x*-axis.
Such intercepts have coordinates *x*,0),*f* (*x*) = 0.**roots** of the function or **zero points** of the function. See also
Bisection and
Newton method in Sequences - Theory - Applications. If you are curious about
*multiplicity of a root*, look at
this note.

- *y*-intercepts are intersections of the graph and the *y*-axis.
Such intercepts have coordinates *f* (0)),*f* (if it is possible). If 0 is not in the domain
of *f*, then there is no *y*-intercept.

This property is very important, not just for drawing graphs. We focus on two
kinds of symmetry, symmetry about the *y*-axis and symmetry about the
origin. One would be also tempted to consider symmetry about the
*x*-axis, but since a function cannot have two values for one *x*,
this does not make sense.

Note that by a **symmetric set** we mean any set *M* satisfying the
condition that if *x*∈*M*,*x*)∈*M*.*K*,*K*)*K*,*K*]*K*, also

Definition(symmetry).

Letfbe a real function defined on a symmetric setM. We say that this function isevenif for everyxfromMone has

f(−x) =f(x).We say that this function is

oddif for everyxfromMone has

f(−x) = −f(x).

Recall that the graph of *f* (−*x*)*f* about the *y*-axis. An even function is therefore such
a function that flipping its graph about the *y*-axis does not change
its shape; in other words, the graph is symmetric about the *y*-axis.

Recall also that the graph of *f* (*x*)*f* about the *x*-axis. An odd function is therefore
such a function that flipping its graph about the *y*-axis gives the
same outcome as flipping its graph about the *x*-axis. This sounds a
bit confusing, so perhaps it is better to slightly change the equality:
*f* (*x*) = −*f* (−*x*).*f* about the *x*-axis and then about the *y*-axis, we get the
original graph. This means that odd functions are exactly those that are
symmetric with respect to the origin.

Is there a function that would be both even and odd? Yes, the function
*f* (*x*) = 0*x*.

**Symmetry and operations.**

Here we have some nice rules. If we add, subtract, multiply or divide two
even functions, then the outcome is even. If we add or subtract two odd
functions, the outcome is odd. If we multiply or divide two odd functions,
then the outcome is even.

The proofs are simple, we will show a proof of the last case, since it may
surprise some readers. Let *f*,*g* be two odd functions defined on
a symmetric set *M*. Let
*h* = *f*⋅*g*.*x*∈*M*

*h*(−*x*) = *f*
(−*x*)⋅*g*(−*x*) = (−*f* (*x*))⋅(−*g*(*x*)) = *f* (*x*)⋅*g*(*x*) = *h*(*x*).

Since *h* makes the sign disappear, it is even.

The product and ratio of two functions, one even and one odd, gives an odd function. But if we add or subtract two functions, one even and one odd, then the outcome is not a symmetric function with the exception when one of the two functions is identically zero.

What about composition? If we compose two symmetric functions, the outcome is symmetric. The kind depends on parity, the only way to get an odd function is by composing two odd functions, all other possibilities (odd-even, even-odd, even-even) lead to an even function.

We start with an informal "definition", the most typical case, to get the right feeling for the notion.

Consider a real function *f* defined on the whole real line. Let
*T* be a positive number. We say that *f* is *periodic* with
*period* *T* if for every real number *x* one has

*f* (*x* + *T* ) = *f* (*x*).

We also say in short that such *f* is *T*-periodic. The formula
says that if we look at some point of the graph of *f* and jump to the
right by *T*, the value must stay the same.

Note that once such a condition is true, then also for all integers *n*
one has
*f* (*x* + *n**T* ) = *f* (*x*).*x* + *T* )*x* (it is also a real number) we get
*f* (*x* + 2*T* ) = *f* (*x* + *T* ).*f* (*x* + 2*T* ) = *f* (*x*).*x* + 2*T* )*x*
and get
*f* (*x* + 3*T* ) = *f* (*x* + 2*T* )*f* (*x* + 3*T* ) = *f* (*x*).*T* without
changing the function value, and we can use it as many times as we want. How
do we get to the left? Using the original equality with
*x* − *T* )*x* we get
*f* (*x*) = *f* (*x* − *T* ),*f* (*x*) = *f* (*x* − 2*T* )

The equation
*f* (*x* + *n**T* ) = *f* (*x*)*T*.

Here is an example of a 2-periodic function.

Note that once we find one period, we have infinitely many of them, since
for any natural number *k*, every *T*-periodic*k**T*-periodic.

*f* (*x* + *n*(*k**T* )) = *f* (*x* + (*n**k*)*T* ) = *f* (*x*).

For instance, in the picture above we can also jump left and right by 4 or by 6
or... without changing the value of *f*. Another way to see periodicity:
If we shift the graph of *f* to the right/left by *T*, the graph
does not change. Yet another way to see periodicity (which is sometimes quite
useful): The shape of the whole graph is determined by one piece of it. This
piece has width *T* and the whole graph is obtained by gluing copies of
this piece one after another. In the picture we marked four such possible
pieces.

Once we find a period, we know that all multiples by positive integers are also periods. But sometimes we also get a period by dividing, in case we overlooked some smaller period. For instance, in the above example we may first notice the period 4 and only then realize that it is in fact just a multiple of the period 2. Is there even a smaller period such that 2 would be an integer multiple of it? Well, it does not seem to be true.

It often helps to find a basic period that gives rise to all other periods,
that is, a period *p* so that all other periods are of the form
*n**p*. Oftentimes it can be done. For instance, one can show that
if we find many periods and all of them are integers, then their greatest
common divisor is also a period. In the above picture, 2 seems to be the
basic period. Sometimes people distinguish by saying that, in the above
example, 4 is *a* period and 2 is *the* period, but it is not
generally accepted; moreover, it is easy to miss such a fine distinction, and
it is not even possible if we use the other way, when we say that *f* is
4-periodic or 2-periodic. Fortunately, it is not a problem, since in
applications we rarely need to know exactly the basic period. Moreover, it
can happen that no such basic period exists.

For instance, it may happen that a function has two dictinct periods and we
are unable to find a common one that would work for both as basic. In some
strange cases things are even stranger, for instance, the constant function
*f* (*x*) = 13*T* > 0*x* one has
*f* (*x* + *T* ) = 13 = *f* (*x*).

Quite often one would like to use the idea of periodicity (the fact that the graph is a repetition of some basic pattern) also for functions whose domain is not the whole real line. The algebraic expression would be just as above, but now one has to be careful not to jump out of the domain with argument. Here comes the definition:

Definition.

Consider a real functionf. LetTbe a positive number. We say thatfisperiodicwith periodTif for every real numberxsuch thatand x∈D(f)( one hasx + T)∈D(f)

f(x+T) =f(x).

As above, from the basic definition we get that also
*f* (*x* + *nT* ) = *f* (*x*)*n*, as long as the arguments in the equation belong to the
domain of *f*. How do you imagine periodic functions now that we have
this general definition? In fact the idea is very similar, the whole graph is
obtained by repeating one piece of it of width *T*. There might be
infinitely many repetitions, just now the domain may contain holes, as you
can see in the following picture. We marked two possible pieces.

But there may also be just a finite number of repetitions, for instance the following function seems to be (1/2)-periodic.

What can be said about **periodic functions and operations**? When we
add/subtract, multiply or divide two periodic functions, the outcome is again
periodic. The period of the outcome is the least common multiple of the two
periods. This statement says that existing periods are preserved, but it can
actually happen that new ones appear. For instance, when we divide sine by
cosine, both

When we compose two functions and the inner function is periodic, then the
whole composition is periodic. Finally, if a function *f* is
*T*-periodic,*f* (*A**x*)*T*/|*A*|-periodic.

Continuity of real functions

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functions