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Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1

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Question: Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } at x = 1.

Options:

  1. Continuous
  2. Not continuous
  3. Depends on the limit
  4. Only left continuous

Correct Answer: Not continuous

Solution: 

The left limit as x approaches 1 is 1, the right limit is 2, and f(1) = 2. Since the left and right limits do not match, f(x) is not continuous at x = 1.

Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1

Practice Questions

Q1
Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } at x = 1.
  1. Continuous
  2. Not continuous
  3. Depends on the limit
  4. Only left continuous

Questions & Step-by-Step Solutions

Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } at x = 1.
  • Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 for x < 1, f(x) = 2 for x = 1, and f(x) = x + 1 for x > 1.
  • Step 2: Calculate the left limit as x approaches 1. This means we look at f(x) when x is just a little less than 1. For x < 1, f(x) = x^2. So, we find the limit: lim (x -> 1-) f(x) = 1^2 = 1.
  • Step 3: Calculate the right limit as x approaches 1. This means we look at f(x) when x is just a little more than 1. For x > 1, f(x) = x + 1. So, we find the limit: lim (x -> 1+) f(x) = 1 + 1 = 2.
  • Step 4: Find the value of the function at x = 1. According to the definition, f(1) = 2.
  • Step 5: Compare the left limit, right limit, and the value of the function at x = 1. The left limit is 1, the right limit is 2, and f(1) is 2.
  • Step 6: Since the left limit (1) does not equal the right limit (2), the function is not continuous at x = 1.
  • Continuity of Functions – Understanding the definition of continuity at a point, which requires that the left limit, right limit, and the function value at that point are all equal.
  • Piecewise Functions – Analyzing functions defined by different expressions over different intervals and how to evaluate limits and function values at the boundaries.
  • Limits – Calculating left-hand and right-hand limits to determine the behavior of a function as it approaches a specific point.
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