1. 10.3: Continuity

2. Definition of Continuity
A function f is continuous at a point x = c if 1.
2. f (c) exists 3.
A function f is continuous on the open interval (a,b) if it is continuous at each point on the interval.
If a function is not continuous, it is discontinuous.

3. A constant function is continuous for all x.
For integer n > 0, f (x) = xn is continuous for all x.
A polynomial function is continuous for all x.
A rational function is continuous for all x, except those values that make the denominator 0.
For n an odd positive integer, is continuous wherever f (x) is continuous.
For n an even positive integer, is continuous wherever f (x) is continuous and nonnegative.

4. Continuous at -1?
Continuous at -2?
YES NO
Continuous at 2?
YES
Continuous at 0?
Continuous at 2?
YES NO
Continuous at -4?
YES

5. Discuss the continuity of each function
(-∞,∞)
This function is continuous on
(-∞,-1)U(-1,3)U(3, ∞) because x can’t equal to -1 or 3
This function is continuous on
(-20,∞) because
X + 20 ≥ 0 so x ≥ - 20
This function is continuous on
(-∞,∞) because
the radicand is always ≥ 0
This function is continuous on
This function is continuous on
(-∞,∞)
(-∞,1)U(1, ∞) because x can’t equal to 1
This function is continuous on