There is no algorithm for finding periodicity, we usually guess and then confirm or rule out our guess. Thus we have two topics here. First, how we confirm/rule out a certain period, and second, how do we make a guess.

**Confirming periodicity.** Assume that we have a function *f* and we
suspect that a certain *T* > 0*x*,
*f* (*x* + *T* ) = *f* (*x*).*x*, then
this *T* is not a period.

Here we will prove that the tangent is actually π-periodic, using some trig identities.

**Guessing periods.** This is based on two ingredients. First, we do know
that some functions are periodic with certain basic periods. In fact, most
people know only four such functions, namely sine and cosine with basic the
period

Second, we need to know what happens to a certain period if we start combining functions together. These facts can come handy:

• If *f* is *T*-periodic, then
*f* (*A**x* + *B*)*T*/|*A*|)-periodic*B* is irrelevant, that is, shift in
argument does not influence periodicity). On the other hand,
*A**f* (*x*) + *B**T*-periodic,*f* do not influence periodicity.

• If *f* is *T*-periodic, then
*f* |*T*-periodic.*x*)*x*)|

On the other hand, *f* (|*x*|)*x*|)

• If *g* is *T*-periodic, then the composition
*f* (*g*)*T*-periodic*f*. Again, note that the basic
period may change, after all,
*x*)|*x*|.*f* periodic does not guarantee anything about the composition.

• If *f* is *T*-periodic and *g* is
*S*-periodic,*f* + *g*,*f* − *g*,*f*⋅*g*,*f* /*g**R*-periodic, where *R* is the least
common multiple of *S* and *T*. Again the basic period may change,
for instance *T* and *S*
need not be integers, so to be precise we add an exact statement:
*R* is the smallest positive number such that
*R*/*T**R*/*S*

**Example:** Determine the periodicity of

**Solution:** We have a sum of two functions, so we start by investigating
each term separately.

Inside the first term we have sine with scaled argument, so by one rule
above, *x*/2)

In the second term inside we have a tangent of a scaled argument, so the
original period
π will be multiplied by 6
according to one rule above. Thus *x*/6)

So we are adding two terms, one

Now we need to check that this is correct:

The proof is complete, our guess was correct. In fact, this is also the basic period, but proving such a thing is not all that easy in general, so we will leave this topic.

For more examples see Solved Problems.

Continuity

Back to Methods Survey - Basic
properties of real functions