ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

1.11 Defining Continuity at a Point

#continuity

#definition

โฑ๏ธย ย 2 min read

written by

Anusha Tekumulla

anusha tekumulla


Conditions for Continuity

๐ŸŽฅWatch: AP Calculus AB/BC - Continuity, Part I This topic is important to the rest of the unit because you will learn what continuity is. In order for a function f(x) to be continuous at a point x = a, it must fulfill ALL THREE of the following conditions.ย โ˜‘๏ธ

  1. f(a) is defined

  2. limit as x approaches a for f(x) exists

  3. limit as x approaches a for f(x) = f(a)

While these might look confusing at first, the conditions are actually quite simple. The first condition checks whether the function has a value at that particular x value. This condition ensures that there isnโ€™t a hole or asymptote at the x value. The second condition checks whether the right-handed and left-handed limits are the same. This condition ensures there isnโ€™t a jump discontinuity at the x value. The last condition checks whether the right-handed and left-handed limits are equal to the actual y value or the function at that particular x value. Thus, the last conditions make sure that there isnโ€™t point discontinuity at the x value.ย ๐Ÿ’ก

As you can see, these three conditions are used to ensure there arenโ€™t any discontinuities at that x value. If any of the conditions fail, there is a discontinuity and the function is not continuous at the x value.ย 

If youโ€™re still confused about how the conditions work, take a look at the example below.ย 

Example Problem: Using Conditions to Determine Continuity

Is the following function continuous at the point x = 3? ๐Ÿง

f(x) = { x + 1, x < 3

ย ย ย ย ย ย ย ย ย 2x - 2, x โ‰ฅ 3 }

ย Condition 1: Does f(3) exist?ย 

Yes! f(3) = 2(3) - 2 = 4

Condition 2: Does the limit as x approaches 2 for f(x) exist?ย 

For this condition, we must check the left and right-handed limits. The left-handed limit is x 3- f(x)= 3 + 1 = 4. The right-handed limit is x 3+ f(x)= 2(3) - 2 = 4. Because the two limits are the same, the limit exists at x = 3.ย 

Condition 3: Does the limit as x approaches 2 for f(x) = f(3)?

Yes! The two equal each other and thus, the function is continuous at x = 3.ย ๐Ÿ™†โ€โ™‚๏ธ๐Ÿ™†

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