The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:

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Q: The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:

Solution: Both conditions must hold true for continuity at x = 2.

Steps: 7

Step 1: Understand what it means for a function to be continuous at a point. A function is continuous at a point if the limit of the function as it approaches that point is equal to the value of the function at that point.
Step 2: Identify the point we are checking for continuity, which is x = 2 in this case.
Step 3: Calculate the limit of f(x) as x approaches 2 from the left (x < 2). This means we use the part of the function f(x) = x^2. So, we find the limit as x approaches 2: limit as x -> 2 of x^2 = 2^2 = 4.
Step 4: Calculate the limit of f(x) as x approaches 2 from the right (x > 2). This means we use the part of the function f(x) = 2x. So, we find the limit as x approaches 2: limit as x -> 2 of 2x = 2*2 = 4.
Step 5: Check the value of the function at x = 2. According to the function definition, f(2) = 4.
Step 6: Compare the limits from both sides and the value of the function at x = 2. We found that limit from the left is 4, limit from the right is 4, and f(2) is also 4.
Step 7: Since both limits are equal to the value of the function at x = 2, we conclude that the function is continuous at x = 2.