The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous a

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

Options:

Correct Answer: No

Solution:

At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.

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

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

Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 for x < 0, f(x) = 2 for x = 0, and f(x) = x + 1 for x > 0.
Step 2: To check if the function is continuous at x = 0, we need to find the left-hand limit (lim x→0-) and the right-hand limit (lim x→0+).
Step 3: Calculate the left-hand limit: lim x→0- f(x) means we look at values of x that are slightly less than 0. For these values, f(x) = x^2. So, lim x→0- f(x) = 0^2 = 0.
Step 4: Calculate the right-hand limit: lim x→0+ f(x) means we look at values of x that are slightly greater than 0. For these values, f(x) = x + 1. So, lim x→0+ f(x) = 0 + 1 = 1.
Step 5: Compare the left-hand limit and the right-hand limit. We found that lim x→0- f(x) = 0 and lim x→0+ f(x) = 1.
Step 6: Since the left-hand limit (0) is not equal to the right-hand limit (1), the function is discontinuous at x = 0.

Piecewise Functions – Understanding how piecewise functions behave at different intervals and how to evaluate limits from both sides.
Continuity – Determining the continuity of a function at a point by checking the limits and the function's value at that point.
Limits – Calculating left-hand and right-hand limits to assess the behavior of a function as it approaches a specific point.