Page 183 - DMTH401_REAL ANALYSIS

Unit 13: Discontinuities and Monotonic Functions

By looking at sequences involving integer multiples of  or /2 we can see that the limit of f(x) Notes
as x approaches zero from the right and from the left both do not exist. Hence, f(x) has an
essential discontinuity at x = 0.

Self Assessment

Fill in the blanks:
1. The class of monotonic functions consists of both the ………………..
2. ……………….. have no discontinuities of second kind.

3. Let f be monotonic on (a; b), then the set of points of (a; b) at which f is ………………… at
most countable.
4. A function f(x) is said to be ……………….. over an interval (a, b) if the derivatives of all
orders of f are nonnegative at all points on the interval.
5. The term ……………….. can also possibly cause some confusion because it refers to a
transformation by a strictly increasing function

13.6 Summary

 If a function fails to be continuous at a point c, then the function is called discontinuous at
c, and c is called a point of discontinuity, or simply a discontinuity.

 In calculus, a function f defined on a subset of the real numbers with real values is called
monotonic (also monotonically increasing, increasing or non-decreasing), if for all x and
y such that x  y one has f(x)  f(y), so f preserves the order (see Figure 1).
 Suppose f is a function with domain D and  D is a point of discontinuity of f.
if limf(x) exists, then c is called removable discontinuity.
x® c
if limf(x) does not exist, but both lim f(x) and lim f(x) exit, then c is called a discontinuity
x® c x® c - x® c +
of the first kind, or jump discontinuity

if either lim f(x) or lim f(x) does not exist, then c is called a discontinuity of the second
x® c - x® c +
kind, or essential discontinuity
If f is a monotone function on an open interval (a, b), then any discontinuity that f may
have in this interval is of the first kind.
If f is a monotone function on an interval [a, b], then f has at most countably many
discontinuities.

13.7 Keywords

Monotonic Transformation: The term monotonic transformation can also possibly cause some
confusion because it refers to a transformation by a strictly increasing function.
Monotonically Decreasing: A function is called monotonically decreasing (also decreasing or
non-increasing) if, whenever x  y, then f(x)  f(y), so it reverses the order.
Monotonic Function: In mathematics, a monotonic function (or monotone function) is a function
that preserves the given order.

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