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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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If $\"\"$ does not satisfies any of these three conditions, then $\"\"$ is said to be discontinuous.

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(a)

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The function is discontinuous at $\"\"$ because the function has a hole at $\"\"$ .

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Therefore $\"\"$ is not defined.

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The function is discontinuous at $\"\"$ because limit does not exist at $\"\"$ .

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$\"\"$ does not exist.

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The function is discontinuous at $\"\"$ because limit does not exist $\"\"$ .

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$\"\"$ does not exist.

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The function is discontinuous at $\"\"$ because limit does not exist $\"\"$ .

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$\"\"$ does not exist.

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(b)

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The function has a hole at $\"\"$ .

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Therefore the function is neither continuous from left side nor from the right side.

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The solid dot indicates the value of the function.

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

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$\"\"$ .

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

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$\"\"$ .

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

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$\"\"$ .

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(a) The function is discontinuous at $\"\"$ .

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(b)

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The function is neither continuous from left side nor from the right side at $\"\"$ .

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

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

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