Basic Calculus - Continuity, Summaries of Mathematics

Download Basic Calculus - Continuity and more Summaries Mathematics in PDF only on Docsity!

CONTINUITY

Answer:

The function is

continuous at

x = 1.

Answer:

The function is

discontinuous

at x = 0.

Answer:

The function is

continuous at

x = 2.

Determine whether is continuous at. Recall: Solution:

𝟐

𝟐

lim

𝒙 → 𝟏

𝟐

+ 𝒙 − 𝟐 )^ = 𝟒 (^ 𝟏 )

𝟐 +(^ 𝟏 )^ − 𝟐 = 𝟑 lim 𝒙 → 𝟏 𝒇 ( 𝒙 )= 𝒇 ( 𝟏 ) ∴ 𝑻𝒉𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒊𝒔 𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔 𝒂𝒕 𝒙 = 𝟏.

The graph of.

Determine whether is continuous at. Solution at x = 2: a) b) c) Recall: The three conditions are satisfied. The function is continuous at x =2.

Determine whether is continuous at. Solution at x = 3: a) indeterminate b) c) Recall: The three conditions are not satisfied. The function is discontinuous at x =

Example 3: Investigate the continuity of the function Solution at x = 3: c) The second condition is not satisfied at x = 3. Hence, the function discontinuous at x = 3. Recall:

The graph of ( − 𝟐 𝒙 + 𝟒 ) 𝒊𝒇 𝒙 ≥ 𝟑 ( 𝒙 − 𝟏 ) 𝒊𝒇 𝒙 < 𝟑

The graph of

• Example

CONTINUITY ON A CLOSED INTERVALCONTINUITY ON A CLOSED INTERVAL A function is said to be continuous ON A CLOSED INTERVAL [a, b] if

Example: Determine whether the following functions is continuous on the given interval.. Example: Determine whether the following functions is continuous on the given interval.. a = -3 and b = 3 Therefore, it is continuous from the right at -3. Therefore, it is continuous from the right at -3. Therefore, it is continuous from the left at 3. Therefore, it is continuous from the left at 3. 2 ND^ condition: 3 rd^ condition: 1 st^ condition: the function is continuous on the open interval (-3, 3) Must be continuous in an open interval (a, b) X Y -2.5 1. -2 2. 0 3 2 2. -2.5 1. Must be continuous from the right at. Must be continuous from the left at.