Download Calculus Notes for 1.4 ***Memorize***Definition of Continuity at a

Calculus Notes for 1.4
***Memorize***Definition of Continuity at a Point:A function f is continuous at c if the
following 3 conditions are met.
1. f(c) is defined. 2. lim f ( x ) exists 3. lim f ( x ) = f(c)
x c
x c
Continuity on an open interval: A function is continuous on an open interval (a,b) if it is
continuous at each point in the interval.
If a function f is defined on an interval I (except possibly at c), and f is not continuous at c, then f
is said to have a discontinuity at c. A discontinuity is said to be removable if f can be made
continuous by appropriately defining (or redefining) f(c). Otherwise, the discontinuity is said to
be nonremovable.
Theorem 1.10 The Existence of a Limit: Let f be a function and let c and L be real numbers.
The limit of f(x) as x approaches c is L if and only if lim f ( x)  L and lim f ( x)  L
x c
x c
A function f is continuous on the closed interval [a,b] if it is continuous on the open interval
(a,b) and lim f ( x)  f (a) and lim f ( x)  f (b)
x b
x a
Definition of Continuity on a Closed Interval
A function is continuous on a closed interval [a,b] if it is continuous on the open interval (a,b)
and
lim+ 𝑓(𝑥) = 𝑓(𝑎)
𝑥→𝑎
and
lim 𝑓(𝑥) = 𝑓(𝑏)
𝑥→𝑏 −
Intermediate Value Theorem: If f is continuous on the closed interval [a,b], f(a) ≠ f(b), and k is
any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.
Properties of Continuity: If b is a real number and f and g are continuous at x = c, then the
following functions are also continuous at c: 1. bf 2. f±g 3. fg 4. f/g
Example 1: Discuss the continuity of each function: this means to 1) tell any x-values where the
function is discontinuous, 2) tell if each discontinuity is removable or non-removable and 3)tell
which reason from the definition of continuous is not met
a) 𝑓(𝑥) =
b) 𝑓(𝑥) =
1
𝑥
𝑥 2 −1
𝑥−1
c) 𝑓(𝑥) = {
𝑥 + 1 𝑓𝑜𝑟 𝑥 ≤ 0
𝑥 2 + 1 𝑓𝑜𝑟 𝑥 > 0
d) y = sin(x)
Example 2: 𝑓(𝑥) = √4 − 𝑥 2 .
Find
lim 𝑓(𝑥)
𝑥→−2+
The Greatest Integer function: ⟦𝑥⟧ = greatest integer n such that n≤x
Example 3: f(x) = ⟦𝑥⟧
lim 𝑓(𝑥)
𝑥→0+
lim 𝑓(𝑥)
𝑥→0−