My Math Forum continuous at any point iff continuous at origin

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 October 22nd, 2008, 01:26 PM #1 Member   Joined: Sep 2008 Posts: 46 Thanks: 0 continuous at any point iff continuous at origin f: A-->B is continuous at any point X iff f is continuous at the origin. (f is a linear map) this does not mean continuity everywhere but rather continuity at any fixed point iff continuity at origin. i'm not sure how to do this formally with epsilons and delta, but if f is continuous at ever point then it must be continuous at the origin. the more important thing is proving the converse. I'm not too sure about this part but perhaps I suppose it isn't continuous at an arbitrary point, then I say that I can construct a sequence such that f is discontinuous at 0, which contradicts my assumption. ... I need help with a formal solution. thanks.
 October 22nd, 2008, 02:54 PM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: continuous at any point iff continuous at origin What conditions are you imposing on f? This isn't true in general; for instance, the function $f\,:\,\mathbb{R}\mapsto\mathbb{R}\\[10pt] f(x)=\begin{cases} 0, & -1\leq x\leq x \\ 1, & x<-1\text{ or }x>1 \end{cases}$ is continuous at 0 but discontinuous at $\pm1.$ We can even find a function that is continuous only at the origin - consider $g\,:\,\mathbb{R}\mapsto\mathbb{R}\\[10pt] g(x)=\begin{cases} x, & \text{if }x\in\mathbb{Q}\\ 0, & \text{otherwise}. \end{cases}$
 October 22nd, 2008, 05:17 PM #3 Member   Joined: Sep 2008 Posts: 46 Thanks: 0 Re: continuous at any point iff continuous at origin Sorry. f is a linear map and it isn't prove it is continuous everywhere iff continuous at origin. instead it is continuous at any fixed point X iff continuous at the origin. does this help?
 October 23rd, 2008, 02:41 AM #4 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 Re: continuous at any point iff continuous at origin T;X->Y X,Y are linear norm space ; the condition of T can be weaken to Tax=aTx(a:C) any(e>0)exst(d>0)all(x:X)|x|<=d ->|Tx| |Tx|0)|Tx|
 October 23rd, 2008, 05:57 AM #5 Member   Joined: Sep 2008 Posts: 46 Thanks: 0 Re: continuous at any point iff continuous at origin thanks. but where are a,C,c coming from? i'm not entirely sure i understand your solution.
 October 23rd, 2008, 04:37 PM #6 Member   Joined: Sep 2008 Posts: 46 Thanks: 0 Re: continuous at any point iff continuous at origin i think i figured a way to prove this. thanks for the help though. your solution seems more terse than mine.
 October 24th, 2008, 12:58 AM #7 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 Re: continuous at any point iff continuous at origin I'm flattered

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