$\"\"$

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The piecewise function is $\"\"$ , $\"\"$ .

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Definition of continuity :

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A function $\"\"$ is continuous at $\"\"$ , if $\"\"$ then it should satisfy three conditions :

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(1) $\"\"$ is defined.

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(2) $\"\"$ exists.

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(3) $\"\"$ .

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$\"\"$

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The piecewise function is $\"\"$ .

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Left hand limit :

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$\"\"$

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$\"\"$ .

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Right hand limit :

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$\"\"$

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$\"\"$ .

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Left hand limit and right hand limit are not equal, so limit does not exist.

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$\"\"$ does not exist.

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It does not satisfies the condition of continuity at $\"\"$ .

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Therefore $\"\"$ is discontinuous at $\"\"$ .

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$\"\"$

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Graph :

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Graph the piecewise function $\"\"$ :

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$\"\"$

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Observe the graph.

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As $\"\"$ approaches to $\"\"$ from left hand side, $\"\"$ tends to $\"\"$ .

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As $\"\"$ approaches to $\"\"$ from right hand side, $\"\"$ tends to $\"\"$ .

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Limit does not exist because the left and right hand limits are not equal.

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The function $\"\"$ is discontinuous at $\"\"$ .

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$\"\"$

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The function $\"\"$ is discontinuous at $\"\"$ .