객관식
Determine if the graph of the function is continuous and/or differentiable at .
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2. Intro to Derivatives
Differentiability
Multiple Choice
Determine if the graph of the function is continuous and/or differentiable at .
A
Continuous and non-differentiable
B
Continuous and differentiable
C
Discontinuous and non-differentiable
D
Discontinuous and differentiable
검증된 단계별 안내1
Step 1: Analyze the graph of the function f(x) at x=1. Observe that there is a jump discontinuity at x=1, as the graph has a gap between the left-hand limit and the right-hand limit.
Step 2: Recall the definition of continuity. A function is continuous at a point if the left-hand limit, right-hand limit, and the value of the function at that point are all equal. Since there is a jump discontinuity at x=1, the function is not continuous at this point.
Step 3: Recall the definition of differentiability. A function is differentiable at a point if it is continuous at that point and has a well-defined derivative. Since the function is not continuous at x=1, it cannot be differentiable at this point.
Step 4: Conclude that the function f(x) is discontinuous and non-differentiable at x=1 based on the analysis of the graph and the definitions of continuity and differentiability.
Step 5: Verify the reasoning by observing the graph again. The jump discontinuity confirms the lack of continuity, and the absence of continuity confirms the lack of differentiability at x=1.
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