Let the function f : R → R be discontinuous at c. Then the statement: ∀ ϵ &gt; 0 , ∃ δ ∈ R , ∀ x ∈ R ( | x − c | &lt; δ ⟹ | f ( x ) − f ( c ) | &lt; ϵ ) is false. The negation of the statement: ∃ ϵ &gt; 0 , ∀ δ ∈ R , ∃ x ∈ R ( | x − c | &lt; δ a n d | f ( x ) − f ( c ) | ⩾ ϵ ) is false because whenever δ is negative, | x − c | &lt; δ is false. Is anything wrong here? Thank you!

Question

Let the function   f  :      R    →      R   be discontinuous at c. Then the statement:   ∀  ϵ  &amp;gt;  0  ,  ∃  δ  ∈      R    ,  ∀  x  ∈      R    (      |    x  −  c      |    &amp;lt;  δ    ⟹        |    f  (  x  )  −  f  (  c  )      |    &amp;lt;  ϵ  ) is false. The negation of the statement:   ∃  ϵ  &amp;gt;  0  ,  ∀  δ  ∈      R    ,  ∃  x  ∈      R    (      |    x  −  c      |    &amp;lt;  δ    a  n  d        |    f  (  x  )  −  f  (  c  )      |    ⩾  ϵ  ) is false because whenever   δ is negative,       |    x  −  c      |    &amp;lt;  δ is false. Is anything wrong here? Thank you!

dkmc4175fl · Accepted Answer

Your initial comment was intended to convey &quot;continuity at&amp;nbsp;c&quot;, but it&#039;s not written precisely enough! In fact, the original statement you gave is true, because you can pick&amp;nbsp;δ&amp;nbsp;=&amp;nbsp;0, regardless of the value of&amp;nbsp;ϵ.The amended version of the original statement is as follows:∀&amp;nbsp;ϵ&amp;nbsp;&amp;gt;&amp;nbsp;0&amp;nbsp;&amp;nbsp;∃&amp;nbsp;δ&amp;nbsp;&amp;gt;&amp;nbsp;0&amp;nbsp;&amp;nbsp;∀&amp;nbsp;x&amp;nbsp;∈&amp;nbsp;R&amp;nbsp;(&amp;nbsp;|&amp;nbsp;x&amp;nbsp;−&amp;nbsp;c&amp;nbsp;|&amp;nbsp;&amp;lt;&amp;nbsp;δ&amp;nbsp;&amp;nbsp;⟹&amp;nbsp;&amp;nbsp;|&amp;nbsp;f&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;−&amp;nbsp;f&amp;nbsp;(&amp;nbsp;c&amp;nbsp;)&amp;nbsp;|&amp;nbsp;&amp;lt;&amp;nbsp;ϵ&amp;nbsp;)This statement is actually false (by definition), assuming that&amp;nbsp;f&amp;nbsp;is not continuous at&amp;nbsp;c. The negation is∃&amp;nbsp;ϵ&amp;nbsp;&amp;gt;&amp;nbsp;0&amp;nbsp;&amp;nbsp;∀&amp;nbsp;δ&amp;nbsp;&amp;gt;&amp;nbsp;0&amp;nbsp;&amp;nbsp;∃&amp;nbsp;x&amp;nbsp;∈&amp;nbsp;R&amp;nbsp;(&amp;nbsp;|&amp;nbsp;x&amp;nbsp;−&amp;nbsp;c&amp;nbsp;|&amp;nbsp;&amp;lt;&amp;nbsp;δ&amp;nbsp;&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;&amp;nbsp;|&amp;nbsp;f&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;−&amp;nbsp;f&amp;nbsp;(&amp;nbsp;c&amp;nbsp;)&amp;nbsp;|&amp;nbsp;≥&amp;nbsp;ϵ&amp;nbsp;)Really, this is accurate.

AA epsilon>0,EEdelta in R,AAx in R(|x−c|<delta=>|f(x)−f(c)|<epsilon) is false?

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Answer & Explanation

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