Use the graph of f(x) to determine if the function is continuous or discontinuous at x=c. c=3
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 = 3. Observe that there is a break in the graph at x = 3, indicating a potential discontinuity.
Step 3: Check if f(3) is defined. From the graph, there is no point at x = 3 on the curve, which means f(3) is not defined.
Step 4: Evaluate the limit of f(x) as x approaches 3. The graph shows that the left-hand limit and right-hand limit do not converge to the same value, indicating the limit does not exist.
Step 5: Conclude that the function is discontinuous at x = 3 because it fails the conditions for continuity: f(3) is not defined, and the limit of f(x) as x approaches 3 does not exist.