Show Function is Continuous...3

[nycmathguy]

Calculus
Section 2.5

I want to do 17 but need the continuity on a half-closed, half-open interval. In fact, can you state all the rules for continuity on the following?

(a, b), [a, b], [a, b), (a, b]

For 17, we use the rule for [a, b). What is that rule?

 

[MathLover1]

correct

 

[MathLover1]

17. do it same way, just note 4 is included

 

[nycmathguy]

17. do it same way, just note 4 is included

What is the continuity on an interval rule for
[a, infinity)?

By 4 is included, do you mean the limit does exists at the point (4,4)?

 

[MathLover1]

What is the continuity on an interval rule for
[a, infinity)?
yes, the limit does exists at the point (4,4) and function is continues after a=4 to infinity

rules:

A function is continuous over an open interval (a,b) if it is continuous at every point in the interval.
A function f(x) is continuous over a closed interval of the form [a,b] if it is continuous at every point in (a,b) and is continuous from the right at a and is continuous from the left at b.
A function f(x) is continuous over an interval of the form (a,b] if it is continuous over (a,b) and is continuous from the left at b. Continuity over other types of intervals are defined in a similar fashion.
Analogously, a function f(x) is continuous over an interval of the form [a,b) if it is continuous over (a,b) and is continuous from the right at a.