Abel's theorem in problems and solutions - School of Mathematics

The complex numbers 63
of continuity, is rather difficult. Hence we give the rigourous definition
of continuity and by means of it we will prove some basic properties of
continuous functions. We give the definition of continuity for functions
of one real variable as well as for functions of one complex variable.
The graph of a function of one real argument can be discontinuous at
some points, and at some points it can have some breaks. It is therefore
natural to consider first the notion of continuity of a function at a given
point rather than the general definition of continuity.
If we try to define more precisely our intuitive idea of continuity of a
function at a given a point we obtain that continuity means the
following: under small changes of the argument near the point the value
of the function changes a little with respect to value Moreover, it
is possible to obtain a variation of the function’s value about as
small as we want by choosing a sufficiently small interval of the variation
of the argument around One can formulate this more rigourously in
the following way.
DEFINITION. Let be a function of one real or complex variable
One says that the function is continuous at if for every arbitrary
real number one can choose a real number (which depends on
and on such that for all numbers satisfying the condition
the inequality 6 holds.
EXAMPLE 15. Prove that the function with complex argument
is continuous at any point Suppose a point and a real number
be given. We have to choose a real number such that for
all numbers which satisfy the condition the inequality
is satisfied. It is not difficult to see that
one can choose (independently of the point Indeed, from the
condition it follows that:
i.e., As a consequence the function is continuous
at any point In particular, it is continuous for all real values of the
argument Therefore if one restricts the function to the real values of
the argument, one obtains that the function of the real variable
is continuous for all real values
6 The geometrical meaning of the inequalities and is
given in Problems 233 and 234.