$\"\"$

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The function is $\"\"$ .

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Find $\"\"$ .

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$\"\"$ is does not exist, because there are infinite number of rational and irrational numbers near to $\"\"$ .

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The function value of these infinite numbers are not approaching a unique value.

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If $\"\"$ is rational the function value is $\"\"$ .

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If $\"\"$ is irrational the function value is $\"\"$ .

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Therefore, $\"\"$ does not exist.

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$\"\"$

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The function is $\"\"$ .

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Find $\"\"$ .

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$\"\"$ definition :

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$\"\"$ means that for every $\"\"$ , there exists a $\"\"$ , such that for every $\"\"$ , the expression $\"\"$ implies $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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$\"\"$ .

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If $\"\"$ is rational, then $\"\"$ .

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$\"\"$

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$\"\"$ .

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If $\"\"$ is irrational, then $\"\"$ .

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$\"\"$ .

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let $\"\"$ .

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$\"\"$ .

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Observe the relationship between two absolute values $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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Therefore $\"\"$ .

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As $\"\"$ tends to $\"\"$ , $\"\"$ approaches to $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$ does not exist.

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$\"\"$ .