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-   -   continuous at any point iff continuous at origin (http://mymathforum.com/real-analysis/4259-continuous-any-point-iff-continuous-origin.html)

 babyRudin October 22nd, 2008 02:26 PM

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.

 mattpi October 22nd, 2008 03:54 PM

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}$

 babyRudin October 22nd, 2008 06:17 PM

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?

 lambda October 23rd, 2008 03:41 AM

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|<e continual on 0
all(x:X)|Tdx/|x||<e <==> |Tx|<e/d|x| ;;;|dx/|x||<=d

exist(c>0)|Tx|<c|x|

so,T:L(X,Y),I even prove that it is

 babyRudin October 23rd, 2008 06:57 AM

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.

 babyRudin October 23rd, 2008 05:37 PM

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.

 lambda October 24th, 2008 01:58 AM

Re: continuous at any point iff continuous at origin

I'm flattered

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