Skip to main content
MathOverflow

Return to Answer

Commonmark migration
Source Link
Community Bot
Community Bot
  • 1
  • 2
  • 3

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that for any $f$ in that set $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that for any $f$ in that set $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that for any $f$ in that set $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.

Improved
Source Link
user30230
user30230

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that for any $f$ in that set $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that for any $f$ in that set $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.

Source Link
user30230
user30230

Existence of such a function goes back to 1939:

J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154

Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that $f^{-1}(\lbrace \alpha \rbrace)$ is perfect except for countably many $\alpha$ and for each of the exceptional $\alpha 's$, $f^{-1}(\lbrace \alpha \rbrace)$ has the form $P\cup\lbrace t\rbrace$ where $P$ is perfect and $t$ is an isolated point of $f^{-1}(\lbrace \alpha \rbrace)$.