1. §2.5. Continuity In this section, we use limits to
formulate the notion of continuity.
This can be treated as one
application of limits.
Continuity at one point
Continuity on an interval
Intermediate value theorem

14. II. Continuity on intervals Definition:
(1) f is continuous on the open interval (a,b) (or (– ,a), (a, ), (– , )) if f is continuous at every number in the interval.
(2) f is continuous on [a,b] if f is continuous at every number in (a,b) and is also continuous from right at a and continuous from left at b

16. Theorem: Polynomials, rational functions, exponential
functions, log functions, root functions, trigonometric
functions, and inverse trigonometric functions are
continuous at every number in their domains.
Comment:
(1) Above functions are called basic functions. So any
basic function is continuous on its domain.
(2) The greatest integer function is not a basic function
because it has discontinuities on its domain.

17. Theorem: (1)Any function created by using composition
from the basic functions is continuous on its domain.
(2) Any function created using combination from
the basic functions is continuous on its domain.
Comment: The greatest integer function can not be
created using basic functions through composition,
or combination because it has discontinuities in its
domain.

20. Comment:
Most functions used in this course are continuous on
their domains. But we still have some functions that
have discontinuities on their domains. For example,
the greatest integer function f(x) = [x].
How to identify discontinuous points:
Usually, there are only a few points in Df where a
discontinuity can occur. A suspicious discontinuity
point c comes from:
(i) The defining rule for f changes at x = c
(ii) substitution of x = c causes division by 0 in f