is not defined at

is defined at ，but does not exist

is defined at ，but

is not defined near

Let ，decide whether following functions are continuous at and why

Let be a point of discontinuity for

If is bounded near ，then it is called removable (In the context of this course)

If both , then it is a pole

If is defined on , and ，then is left continuous at

If is defined on ，and ，then is right continuous at

If is defined in , then it is continuous at iff it is both left and right continuous at

If is continuous at every point of an open interval , then we say that is continuous on

If is continuous on , and：
right continuous at ，then we say that is continuous on
left continuous at ，then we say that is continuous on
continuous on and ，then we say that is continuous on

Continuity is preserved upon finite additions and multiplications

Continuity is preserved upon composition

Continuity is not necessarily preserved upon inversion

If is continuous，then attains both maximal and minmal values on

i.e., there exist so that hold for all

Counterexample： has no maximum or minimum on
Counterexample：： has no maximum or minimum on

If is continuous，with maximum and minimum ，then assumes all values between and

i.e., for any ，there exists so that

In particular if ，then has at least one root in

If is continuous，then for any ，there exist ，so that

Denote the maximum and the minimum of on by respectively

Clearly