The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous a

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Question: The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?

Options:

Correct Answer: No

Solution:

Left limit as x approaches 1 is 2, right limit is 1, but f(1) = 2. Hence, it is discontinuous at x = 1.

The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?

The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?

Step 1: Identify the function f(x) and its pieces: f(x) = { x + 1 for x < 1; 2 for x = 1; x^2 for x > 1 }.
Step 2: Find the left limit as x approaches 1. This means we look at f(x) when x is just less than 1. So, we use the piece x + 1.
Step 3: Calculate the left limit: f(1-) = 1 + 1 = 2.
Step 4: Find the right limit as x approaches 1. This means we look at f(x) when x is just greater than 1. So, we use the piece x^2.
Step 5: Calculate the right limit: f(1+) = 1^2 = 1.
Step 6: Check the value of the function at x = 1. f(1) = 2.
Step 7: Compare the left limit, right limit, and the value of the function at x = 1. Left limit is 2, right limit is 1, and f(1) is 2.
Step 8: Since the left limit (2) does not equal the right limit (1), the function is discontinuous at x = 1.

Piecewise Functions – Understanding how piecewise functions behave at specific points, particularly at the boundaries of their defined intervals.
Limits and Continuity – Evaluating the left-hand limit, right-hand limit, and the function value at a point to determine continuity.