Question: Is the function f(x) = |x| continuous at x = 0?
Options:
Yes
No
Only left continuous
Only right continuous
Correct Answer: Yes
Solution:
The function f(x) = |x| is continuous at x = 0 because the left limit, right limit, and f(0) all equal 0.
Is the function f(x) = |x| continuous at x = 0?
Practice Questions
Q1
Is the function f(x) = |x| continuous at x = 0?
Yes
No
Only left continuous
Only right continuous
Questions & Step-by-Step Solutions
Is the function f(x) = |x| continuous at x = 0?
Step 1: Understand what continuity means. A function is continuous at a point if the left limit, right limit, and the function value at that point are all the same.
Step 2: Identify the function we are examining, which is f(x) = |x|.
Step 3: Find the value of the function at x = 0. Calculate f(0) = |0| = 0.
Step 4: Calculate the left limit as x approaches 0. This means looking at values of x that are slightly less than 0. The left limit is lim (x -> 0-) f(x) = lim (x -> 0-) |x| = 0.
Step 5: Calculate the right limit as x approaches 0. This means looking at values of x that are slightly greater than 0. The right limit is lim (x -> 0+) f(x) = lim (x -> 0+) |x| = 0.
Step 6: Compare the left limit, right limit, and the function value at x = 0. We found that left limit = 0, right limit = 0, and f(0) = 0.
Step 7: Since all three values are equal (0), we conclude that the function f(x) = |x| is continuous at x = 0.
Continuity – Understanding the definition of continuity at a point, which requires that the left limit, right limit, and the function value at that point are all equal.
Absolute Value Function – Recognizing the properties of the absolute value function, particularly its behavior around zero.
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