1.11 defining continuity at a point answer key

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04-02-2025

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This guide delves into the crucial concept of continuity at a point in calculus, providing a clear understanding of the definition and its implications. We'll explore the three conditions that must be met for a function to be continuous at a specific point, illustrated with examples and explanations to solidify your comprehension. This detailed explanation aims to provide a robust answer key beyond a simple yes/no response, equipping you with the knowledge to confidently tackle continuity problems.

Understanding Continuity at a Point

In simpler terms, a function is continuous at a point if you can draw its graph without lifting your pen. However, a rigorous mathematical definition provides more precision. A function f(x) is continuous at a point x = c if and only if the following three conditions are met:

1. f(c) is defined: The function must have a defined value at the point x = c. In other words, c must be within the domain of the function.

2. The limit of f(x) as x approaches c exists: This means that the left-hand limit and the right-hand limit at x = c must be equal. Formally:

lim_x→c^- f(x) = lim_x→c⁺ f(x) = L

where L is a finite number.

3. The limit equals the function value: The limit of the function as x approaches c must equal the function's value at c. That is:

lim_x→c f(x) = f(c)

If even one of these conditions is not satisfied, the function is discontinuous at x = c.

Examples to Illustrate Continuity

Let's examine some examples to solidify our understanding.

Example 1: Continuous Function

Consider the function f(x) = x². Let's check for continuity at x = 2.

1. f(2) = 2² = 4 (defined)
2. lim_x→2 x² = 4 (the limit exists)
3. lim_x→2 x² = f(2) = 4 (limit equals function value)

All three conditions are met; therefore, f(x) = x² is continuous at x = 2. In fact, this function is continuous everywhere.

Example 2: Discontinuous Function (Removable Discontinuity)

Consider the function:

f(x) = (x² - 4) / (x - 2) for x ≠ 2 f(2) = 5

Let's check for continuity at x = 2.

1. f(2) = 5 (defined)
2. lim_x→2 (x² - 4) / (x - 2) = lim_x→2 (x + 2) = 4 (the limit exists)
3. lim_x→2 (x² - 4) / (x - 2) ≠ f(2) (limit does not equal function value)

Since the third condition is not met, f(x) is discontinuous at x = 2. This is a removable discontinuity because we could redefine f(2) = 4 to make the function continuous.

Example 3: Discontinuous Function (Jump Discontinuity)

Consider the piecewise function:

f(x) = x, x < 2 f(x) = x + 1, x ≥ 2

Let's check continuity at x = 2.

1. f(2) = 3 (defined)
2. lim_x→2^- f(x) = 2 and lim_x→2⁺ f(x) = 3 (the limit does not exist)

Since the second condition is not met (the left and right limits are not equal), f(x) is discontinuous at x = 2. This type of discontinuity is known as a jump discontinuity.

Conclusion

Understanding the three conditions for continuity at a point is essential for mastering calculus. By carefully examining each condition and applying the definitions to various examples, you can effectively determine whether a function is continuous at a given point and identify the type of discontinuity if it exists. Remember to always check all three conditions to arrive at the correct conclusion.