Determine if the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 } is co

Determine if the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 } is continuous at x = 1.

Determine if the function f(x) = { x^2, x < 1; 3, x = 1; 2x, x > 1 } is continuous at x = 1.

Correct Answer: Not continuous

Step 1: Identify the function f(x) and the point of interest, which is x = 1.
Step 2: Find the value of the function at x = 1. This is f(1) = 3.
Step 3: Calculate the left-hand limit as x approaches 1 from the left (x < 1). This means using the part of the function for x < 1, which is f(x) = x^2. So, lim x->1- f(x) = 1^2 = 1.
Step 4: Calculate the right-hand limit as x approaches 1 from the right (x > 1). This means using the part of the function for x > 1, which is f(x) = 2x. So, lim x->1+ f(x) = 2*1 = 2.
Step 5: Compare the value of the function at x = 1 (which is 3) with the left-hand limit (1) and the right-hand limit (2).
Step 6: Since f(1) = 3 is not equal to the left-hand limit (1) and the right-hand 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 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 based on the input value, particularly at the point where the definition changes.
Limits – Calculating one-sided limits to determine the behavior of a function as it approaches a specific point.