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in Continuity and Differentiability by (48.6k points)
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Prove that \(f(x) = \begin{cases}(1/2)(x - |x|), & \quad \text{when } x ≠ 0\text{}\\ 2, & \quad \text{when } x = 0 \text{} \end{cases} \) is discontinuous at x = 0.

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It is given that

\(f(x) = \begin{cases}(1/2)(x - |x|), & \quad \text{when } x ≠ 0\text{}\\ 2, & \quad \text{when } x = 0 \text{} \end{cases} \)

Consider left hand limit at x = 0

Here the value of function at x = 0 is f(x) = 2

We get f(0) = 2

So

Therefore, f(x) is discontinuous at x = 0.

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