Graph of continuous functions












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Prove or disprove: There exists a family of continuous functions $f_n:[0,1]rightarrow mathbb{R}$, $nin mathbb{N}$ with graphs $T(f_n)$ such that $[0,1]times mathbb{R}=cup_{nin mathbb{N}}T(f_n)$.



I claim that there is no such a family of functions. By contradiction suppose there exists such a family of functions. So ${0}times mathbb{R}=cup_{nin mathbb{N}} (0,f_n(0))$. So $mathbb{R}$ is a countable unions of real numbers. Contradiction.



However, I didn't use the assumption of continuity in the problem. I'm doubtful my solution is correct.






real-analysis limits functions continuity







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  • 4




    $begingroup$
    Your argument is correct and continuity is not required.
    $endgroup$
    – Kavi Rama Murthy
    Feb 1 at 0:05
















3












$begingroup$


Prove or disprove: There exists a family of continuous functions $f_n:[0,1]rightarrow mathbb{R}$, $nin mathbb{N}$ with graphs $T(f_n)$ such that $[0,1]times mathbb{R}=cup_{nin mathbb{N}}T(f_n)$.



I claim that there is no such a family of functions. By contradiction suppose there exists such a family of functions. So ${0}times mathbb{R}=cup_{nin mathbb{N}} (0,f_n(0))$. So $mathbb{R}$ is a countable unions of real numbers. Contradiction.



However, I didn't use the assumption of continuity in the problem. I'm doubtful my solution is correct.






real-analysis limits functions continuity







share|cite|improve this question









$endgroup$








  • 4




    $begingroup$
    Your argument is correct and continuity is not required.
    $endgroup$
    – Kavi Rama Murthy
    Feb 1 at 0:05














3












3








3





$begingroup$


Prove or disprove: There exists a family of continuous functions $f_n:[0,1]rightarrow mathbb{R}$, $nin mathbb{N}$ with graphs $T(f_n)$ such that $[0,1]times mathbb{R}=cup_{nin mathbb{N}}T(f_n)$.



I claim that there is no such a family of functions. By contradiction suppose there exists such a family of functions. So ${0}times mathbb{R}=cup_{nin mathbb{N}} (0,f_n(0))$. So $mathbb{R}$ is a countable unions of real numbers. Contradiction.



However, I didn't use the assumption of continuity in the problem. I'm doubtful my solution is correct.






real-analysis limits functions continuity







share|cite|improve this question









$endgroup$




Prove or disprove: There exists a family of continuous functions $f_n:[0,1]rightarrow mathbb{R}$, $nin mathbb{N}$ with graphs $T(f_n)$ such that $[0,1]times mathbb{R}=cup_{nin mathbb{N}}T(f_n)$.



I claim that there is no such a family of functions. By contradiction suppose there exists such a family of functions. So ${0}times mathbb{R}=cup_{nin mathbb{N}} (0,f_n(0))$. So $mathbb{R}$ is a countable unions of real numbers. Contradiction.



However, I didn't use the assumption of continuity in the problem. I'm doubtful my solution is correct.






real-analysis limits functions continuity




real-analysis limits functions continuity






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 31 at 23:51









S_AlexS_Alex

20219




20219








  • 4




    $begingroup$
    Your argument is correct and continuity is not required.
    $endgroup$
    – Kavi Rama Murthy
    Feb 1 at 0:05














  • 4




    $begingroup$
    Your argument is correct and continuity is not required.
    $endgroup$
    – Kavi Rama Murthy
    Feb 1 at 0:05








4




4




$begingroup$
Your argument is correct and continuity is not required.
$endgroup$
– Kavi Rama Murthy
Feb 1 at 0:05




$begingroup$
Your argument is correct and continuity is not required.
$endgroup$
– Kavi Rama Murthy
Feb 1 at 0:05










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