Checking for discontinuity of $f(x)= sqrt{(1+p^2)/(1+q^2)}$ if $x = p/q neq 0$, $f(x) = x$ otherwise.











up vote
-2
down vote

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Let $f$ maps from $mathbb{R}$ to $mathbb{R}$. Be given by



$$f(x)=
begin{cases}
sqrt{(1+p^2)/(1+q^2)},& x=p/q neq 0,\
x, & text{otherwise}.
end{cases}
$$
Then



a) $f$ is continuous on $big((0,infty) cap mathbb{Q'}big) cup {0,1}$. (Note that $mathbb{Q}'$ denote $mathbb{Q}$ complement).



b) If $a in (-infty,0)$ then $lim_{xto a} f(x)$ does not exists.



c) If $a in [0,infty)$, then $lim_{xto a} f(x)=a$



I know it is discontinuous, and in the answer sheet a, b, c are correct options. But I confused about option (b), how limit doesn't exist for negative number and also confused about option (a), how it is continuous on 0. I have tried a lot, eventually leads to headache, please help me, I shall be very thankful for it. Thanks in advance.






calculus continuity







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  • I have edited the formatting, let me know if the question is still accurate. Also, when you define $f(x)$ for $x = p/q$, are you assuming that $p/q$ is a rational with no common factors between $p$ and $q$?
    – Brahadeesh
    23 hours ago










  • No such thing is mentioned, this question belongs to a test series of csir net
    – user532616
    13 hours ago










  • Hmm, that’s odd...
    – Brahadeesh
    12 hours ago















up vote
-2
down vote

favorite












Let $f$ maps from $mathbb{R}$ to $mathbb{R}$. Be given by



$$f(x)=
begin{cases}
sqrt{(1+p^2)/(1+q^2)},& x=p/q neq 0,\
x, & text{otherwise}.
end{cases}
$$
Then



a) $f$ is continuous on $big((0,infty) cap mathbb{Q'}big) cup {0,1}$. (Note that $mathbb{Q}'$ denote $mathbb{Q}$ complement).



b) If $a in (-infty,0)$ then $lim_{xto a} f(x)$ does not exists.



c) If $a in [0,infty)$, then $lim_{xto a} f(x)=a$



I know it is discontinuous, and in the answer sheet a, b, c are correct options. But I confused about option (b), how limit doesn't exist for negative number and also confused about option (a), how it is continuous on 0. I have tried a lot, eventually leads to headache, please help me, I shall be very thankful for it. Thanks in advance.






calculus continuity







share|cite|improve this question
























  • I have edited the formatting, let me know if the question is still accurate. Also, when you define $f(x)$ for $x = p/q$, are you assuming that $p/q$ is a rational with no common factors between $p$ and $q$?
    – Brahadeesh
    23 hours ago










  • No such thing is mentioned, this question belongs to a test series of csir net
    – user532616
    13 hours ago










  • Hmm, that’s odd...
    – Brahadeesh
    12 hours ago













up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











Let $f$ maps from $mathbb{R}$ to $mathbb{R}$. Be given by



$$f(x)=
begin{cases}
sqrt{(1+p^2)/(1+q^2)},& x=p/q neq 0,\
x, & text{otherwise}.
end{cases}
$$
Then



a) $f$ is continuous on $big((0,infty) cap mathbb{Q'}big) cup {0,1}$. (Note that $mathbb{Q}'$ denote $mathbb{Q}$ complement).



b) If $a in (-infty,0)$ then $lim_{xto a} f(x)$ does not exists.



c) If $a in [0,infty)$, then $lim_{xto a} f(x)=a$



I know it is discontinuous, and in the answer sheet a, b, c are correct options. But I confused about option (b), how limit doesn't exist for negative number and also confused about option (a), how it is continuous on 0. I have tried a lot, eventually leads to headache, please help me, I shall be very thankful for it. Thanks in advance.






calculus continuity







share|cite|improve this question















Let $f$ maps from $mathbb{R}$ to $mathbb{R}$. Be given by



$$f(x)=
begin{cases}
sqrt{(1+p^2)/(1+q^2)},& x=p/q neq 0,\
x, & text{otherwise}.
end{cases}
$$
Then



a) $f$ is continuous on $big((0,infty) cap mathbb{Q'}big) cup {0,1}$. (Note that $mathbb{Q}'$ denote $mathbb{Q}$ complement).



b) If $a in (-infty,0)$ then $lim_{xto a} f(x)$ does not exists.



c) If $a in [0,infty)$, then $lim_{xto a} f(x)=a$



I know it is discontinuous, and in the answer sheet a, b, c are correct options. But I confused about option (b), how limit doesn't exist for negative number and also confused about option (a), how it is continuous on 0. I have tried a lot, eventually leads to headache, please help me, I shall be very thankful for it. Thanks in advance.






calculus continuity




calculus continuity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 23 hours ago









Brahadeesh

5,53941956




5,53941956










asked 2 days ago









user532616

144




144












  • I have edited the formatting, let me know if the question is still accurate. Also, when you define $f(x)$ for $x = p/q$, are you assuming that $p/q$ is a rational with no common factors between $p$ and $q$?
    – Brahadeesh
    23 hours ago










  • No such thing is mentioned, this question belongs to a test series of csir net
    – user532616
    13 hours ago










  • Hmm, that’s odd...
    – Brahadeesh
    12 hours ago


















  • I have edited the formatting, let me know if the question is still accurate. Also, when you define $f(x)$ for $x = p/q$, are you assuming that $p/q$ is a rational with no common factors between $p$ and $q$?
    – Brahadeesh
    23 hours ago










  • No such thing is mentioned, this question belongs to a test series of csir net
    – user532616
    13 hours ago










  • Hmm, that’s odd...
    – Brahadeesh
    12 hours ago
















I have edited the formatting, let me know if the question is still accurate. Also, when you define $f(x)$ for $x = p/q$, are you assuming that $p/q$ is a rational with no common factors between $p$ and $q$?
– Brahadeesh
23 hours ago




I have edited the formatting, let me know if the question is still accurate. Also, when you define $f(x)$ for $x = p/q$, are you assuming that $p/q$ is a rational with no common factors between $p$ and $q$?
– Brahadeesh
23 hours ago












No such thing is mentioned, this question belongs to a test series of csir net
– user532616
13 hours ago




No such thing is mentioned, this question belongs to a test series of csir net
– user532616
13 hours ago












Hmm, that’s odd...
– Brahadeesh
12 hours ago




Hmm, that’s odd...
– Brahadeesh
12 hours ago















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