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Stardust*
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f uniformly continuous --> finite slope towards infinity
Given [tex]f:R \rightarrow R[/tex] uniformly continuous. Show that [tex]\limsup_{x\rightarrow \infty} \displaystyle|f(x)|/x<\infty[/tex] i.e.
[tex]\exists C \in R: \, |f(x)|\leq C|x| [/tex] as [tex]x \rightarrow \pm \infty[/tex].
The idea of uniform continuity:
[tex]\forall \varepsilon >0, \, \exists \delta>0:\, \forall x,y \in R, |x-y|<\delta \, \Rightarrow \, |f(x)-f(y)|<\varepsilon [/tex].
If f were a linear function, it could be quite easy...
But f is unknown, so I tried sth else:
let's say uniform continuity gives the certainty we can always find a box in which the studied part of the function's graph is contained properly.
In this case we need a corner of the box to be in 0 (as the denominator simply has |x-0|=|x|) and let the other corner move as far as possible towards infinity, along the x-axis. Moreover in vertical we have to forget the the typical |f(x)-f(y)|, leaving |f(x)| only.
This suggests that the contribution of f(y) must somehow be negligible, I mean we can find a larger side of length f(x) for the box to keep the function inside it, without caring about f(y).
These are my thoughts till now. The first problem is: are these ideas right?
The second one is: how to express them in a formal, mathematical language?
Homework Statement
Given [tex]f:R \rightarrow R[/tex] uniformly continuous. Show that [tex]\limsup_{x\rightarrow \infty} \displaystyle|f(x)|/x<\infty[/tex] i.e.
[tex]\exists C \in R: \, |f(x)|\leq C|x| [/tex] as [tex]x \rightarrow \pm \infty[/tex].
Homework Equations
The idea of uniform continuity:
[tex]\forall \varepsilon >0, \, \exists \delta>0:\, \forall x,y \in R, |x-y|<\delta \, \Rightarrow \, |f(x)-f(y)|<\varepsilon [/tex].
The Attempt at a Solution
If f were a linear function, it could be quite easy...
But f is unknown, so I tried sth else:
let's say uniform continuity gives the certainty we can always find a box in which the studied part of the function's graph is contained properly.
In this case we need a corner of the box to be in 0 (as the denominator simply has |x-0|=|x|) and let the other corner move as far as possible towards infinity, along the x-axis. Moreover in vertical we have to forget the the typical |f(x)-f(y)|, leaving |f(x)| only.
This suggests that the contribution of f(y) must somehow be negligible, I mean we can find a larger side of length f(x) for the box to keep the function inside it, without caring about f(y).
These are my thoughts till now. The first problem is: are these ideas right?
The second one is: how to express them in a formal, mathematical language?