If f(x) = x^2 for x < 1 and f(x) = 3 for x ≥ 1, is f(x) continuous at x = 1?

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Question: If f(x) = x^2 for x < 1 and f(x) = 3 for x ≥ 1, is f(x) continuous at x = 1?

Options:

Correct Answer: No

Solution:

At x = 1, f(1) = 3 and limit from left is 1^2 = 1. Since they are not equal, f(x) is discontinuous at x = 1.

If f(x) = x^2 for x < 1 and f(x) = 3 for x ≥ 1, is f(x) continuous at x = 1?

If f(x) = x^2 for x < 1 and f(x) = 3 for x ≥ 1, is f(x) continuous at x = 1?

Step 1: Identify the function f(x). It is defined as f(x) = x^2 for x < 1 and f(x) = 3 for x ≥ 1.
Step 2: Find the value of f(1). Since 1 is greater than or equal to 1, we use the second part of the function: f(1) = 3.
Step 3: Calculate the limit of f(x) as x approaches 1 from the left (x < 1). We use the first part of the function: limit as x approaches 1 from the left is f(x) = x^2 = 1^2 = 1.
Step 4: Compare the value of f(1) and the limit from the left. We found f(1) = 3 and the limit from the left = 1.
Step 5: Since f(1) (which is 3) is not equal to the limit from the left (which is 1), the function f(x) is discontinuous at x = 1.

Continuity of Functions – Understanding the definition of continuity at a point, which requires that the limit from the left equals the limit from the right and both equal the function's value at that point.
Piecewise Functions – Analyzing functions defined by different expressions over different intervals and how to evaluate them at specific points.
Limits – Calculating the limit of a function as it approaches a specific point from both sides.