For which value of c is the function f(x) = { x^2, x < 1; c, x = 1; 2x, x >

For which value of c is the function f(x) = { x^2, x < 1; c, x = 1; 2x, x > 1 } continuous at x = 1? (2022)

For which value of c is the function f(x) = { x^2, x < 1; c, x = 1; 2x, x > 1 } continuous at x = 1? (2022)

Step 1: Understand that a function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal.
Step 2: Identify the function f(x) and the point of interest, which is x = 1.
Step 3: Calculate the left-hand limit as x approaches 1 from the left (x < 1). For f(x), this is x^2. So, find the limit: lim (x -> 1-) f(x) = 1^2 = 1.
Step 4: Calculate the right-hand limit as x approaches 1 from the right (x > 1). For f(x), this is 2x. So, find the limit: lim (x -> 1+) f(x) = 2 * 1 = 2.
Step 5: Set the value of the function at x = 1, which is c, equal to the left-hand limit to ensure continuity. We have c = 1.
Step 6: Check if c also matches the right-hand limit. Since c = 1 does not equal the right-hand limit (which is 2), we need to adjust c.
Step 7: To make the function continuous at x = 1, we need c to equal the left-hand limit, which is 1. Thus, c must be 1.

Continuity of Functions – Understanding the conditions under which a piecewise function is continuous at a specific point.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value.
Limit and Value Matching – Ensuring that the limit of the function as it approaches a point matches the function's value at that point.