Use graph of f(x) to determine if the function is continuous or discontinuous at x=c c=0
A
Continuous
B
Discontinuous
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1
Step 1: Recall the definition of continuity. A function is continuous at a point x = c if the following three conditions are satisfied: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c).
Step 2: Analyze the graph at x = 0. Observe the behavior of the function f(x) near x = 0. Check if there is a break, jump, or hole in the graph at this point.
Step 3: From the graph, note that there is a hole at x = 0, indicating that f(0) is not defined. This violates the first condition for continuity.
Step 4: Additionally, observe that the left-hand limit and right-hand limit of f(x) as x approaches 0 are not equal. This violates the second condition for continuity.
Step 5: Conclude that the function f(x) is discontinuous at x = 0 because it fails to meet the conditions for continuity.