$\"\"$

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The function is $\"\"$ .

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The function $\"\"$ is continuous over the interval $\"\"$ , $\"\"$ and $\"\"$ .

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Check the continuity of the function at $\"\"$ .

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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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Check the continuity of the function at $\"\"$ .

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

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If $\"\"$ then the function is $\"\"$ .

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$\"\"$

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$\"\"$ .

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Condition (2): $\"\"$ exists.

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Left hand limit :

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$\"\"$

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Right hand limit :

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$\"\"$

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

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$\"\"$ does not exist.

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The limit does not exist at $\"\"$ .

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

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$\"\"$ and $\"\"$ .

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$\"\"$ .

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Therefore the function $\"\"$ is continuous at $\"\"$ from left side.

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$\"\"$

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Check the continuity of the function at $\"\"$ .

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

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If $\"\"$ then the function is $\"\"$ .

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$\"\"$

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$\"\"$ .

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Condition (2): $\"\"$ exists.

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Left hand limit :

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$\"\"$

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Right hand limit :

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$\"\"$

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

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$\"\"$ does not exist.

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The limit does not exist at $\"\"$ .

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

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$\"\"$ and $\"\"$ .

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$\"\"$ .

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Therefore the function $\"\"$ is continuous at $\"\"$ from right side.

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$\"\"$

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Graph :

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Graph the function $\"\"$ :

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$\"\"$

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$\"\"$

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The function $\"\"$ is discontinuous at $\"\"$ and it is continuous from left side.

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The function $\"\"$ is discontinuous at $\"\"$ and it is continuous from right side.

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Graph of the function $\"\"$ is

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$\"\"$