# Mathematical analysis: Continuity of functions

Good Evening everyone, I just wrote my final examination for analysis. One of the questions was one that we had in a previous quiz, the solution of which I still don't quite understand. The question is as see below. Could someone please elucidate answer. I don't know how, but there was some confusion in the forum by the wording so, I state clearly: As you can see in the picture. the answer is C). I understand why III) is wrong and II) is correct. but by the definition of a limit: $$for\, every \, \varepsilon > 0 \, there\, exists \, \delta >0 \, such \, that$$ $$0<|x-a| < \delta\, \rightarrow |f(x) - L| <\varepsilon$$ we then have that $$-\varepsilon < f(x) - L < \varepsilon \rightarrow L-\varepsilon and thus f(x) is in $$(L-\varepsilon , L+\varepsilon)$$

so I don't get why I) is wrong.

Thank you.

• Fir (I) and (II) read the definitions of limit and of continuity; for (III) ask yourself where will $g(x)$ be close to as $x$ is close to $c$ and see if that fits the definition. – Will M. Nov 6 '18 at 16:49
• I have edited your post to show the picture directly. – xbh Nov 6 '18 at 17:15
• What are your thoughts? How was it explained after "the previous quiz" (usually any questions students had on a quiz are explained after the result are announced)? In other words, please ask a more concrete question about what in particular you are struggling with – Yuriy S Nov 6 '18 at 18:13
• This looks like an upload of a quiz, not a final exam. Unless you provide more information about your answer, and your thinking and reasoning, how are we to know that you aren't trying to get us to help you with a take-home quiz of some sort? – Namaste Nov 6 '18 at 21:33
• @amWhy, There is an answer on the picture! Asking for help wouldn't be required. but I have edited the question to be as specific as possible. Yuriy S, this memo was uploaded after the quiz. no explanations given. – Destro Nov 8 '18 at 9:29

What is wrong with I) is that the interval for $$x$$ includes the case $$x=a$$ whereas the limit definition excludes $$x=a$$. Why is this important? Consider the case $$f(x)=sin(x)/x$$ and $$L=0$$.