University Mathematics Handbook (2015)

IV. Single-Variable Differential Calculus

Chapter 3. Continuity of Functions

Function is continuous at point if it is defined in the neighborhood of this point, there exists the limit , and it equals to the value of the function at point . That is, .

3.1  Properties of Continuous Functions

a.  If an elementary function is continuous at point , then it is continuous at .

b.  If functions and are defined at point , then:

1.  For every constant , function is continuous at .

2.   is continuous at .

3.   is continuous at .

4.  If, in addition, , then is continuous at .

c.  The composition of two continuous functions is a continuous function.

d.  Let be a function continuous on a closed interval . If (that is, ) attains values of opposite signs at the ends points of the interval, then there exists a point , such that .

e.  For all continuous function , there exists a point , such that . It is called stationary point.

f.  Cauchy's intermediate value theorem: If function is continuous in interval , and real number is between and , then there exists a point , , such that .

g.  Weierstrass extreme value theorem: If function is continuous in finite, closed interval , then is bounded in that interval, and attains its maximum and minimum in that interval. That is, there exists an in the interval such that , for all , and there exists an in the interval such that for all .

3.2  Types of Discontinuities

a.  Let be a function defined in the neighborhood of , except, possibly, . We say that is removable discontinuity if:

1.  There exists a limit .

2.  , or the function is undefined at .

Example: For function , is a removable discontinuity point since there exists , and the function is undefined at .

b.  Point is called first type or jump discontinuity of , if:

1.   is defined in a specific neighborhood of , except, possibly, .

2.  It has the one-sided, finite limits , `.

3.  .

Example: For function , is a jump discontinuity since is defined in the neighborhood of (except in ), and the two one-sided limits , exist and they are different.

c.   is called second type discontinuity of if:

1.   is defined in the neighborhood of , except,
possibly, .

2.  At list one of the one-sided limits at does not exists or in not finite.

Examples:

1.  For function has a second-type discontinuity at , since .

2.  For function , is a second-type discontinuity. does not exist.

3.3  Uniform Continuity

Let function be defined above interval . We say that is uniformly continuous in , if, for every there exists such that for all for which holds , there holds

.

Cantor theorem: A function continuous at closed interval , is uniformly continuous in .