For the function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 }, is f(x) co

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Question: For the function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 }, is f(x) continuous at x = 0?

Options:

Correct Answer: No

Solution:

The left limit as x approaches 0 is 0, the right limit is 1, and f(0) = 0. Since the limits do not match, f(x) is discontinuous at x = 0.

For the function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 }, is f(x) continuous at x = 0?

For the function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 }, is f(x) continuous at x = 0?

Step 1: Identify the function f(x) and its pieces: f(x) = x^2 for x < 0, f(x) = 0 for x = 0, and f(x) = x + 1 for x > 0.
Step 2: Find the left limit as x approaches 0. This means we look at f(x) when x is slightly less than 0. For x < 0, f(x) = x^2. So, as x approaches 0 from the left, f(x) approaches 0.
Step 3: Find the right limit as x approaches 0. This means we look at f(x) when x is slightly more than 0. For x > 0, f(x) = x + 1. So, as x approaches 0 from the right, f(x) approaches 1.
Step 4: Check the value of f(0). According to the function, when x = 0, f(0) = 0.
Step 5: Compare the left limit, right limit, and f(0). The left limit is 0, the right limit is 1, and f(0) is 0.
Step 6: Since the left limit (0) does not equal the right limit (1), f(x) is discontinuous at x = 0.

Continuity of Functions – Understanding the definition of continuity at a point, which requires that the left-hand limit, right-hand limit, and the function value at that point are all equal.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly at the boundaries.
Limit Evaluation – Calculating left-hand and right-hand limits to determine the behavior of a function as it approaches a specific point.