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- - **continuous at any point iff continuous at origin**
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continuous at any point iff continuous at originf: 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. |

Re: continuous at any point iff continuous at originWhat conditions are you imposing on f? This isn't true in general; for instance, the functionis continuous at 0 but discontinuous at We can even find a function that is continuous only at the origin - consider |

Re: continuous at any point iff continuous at originSorry. 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? |

Re: continuous at any point iff continuous at originT;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 |

Re: continuous at any point iff continuous at originthanks. but where are a,C,c coming from? i'm not entirely sure i understand your solution. |

Re: continuous at any point iff continuous at origini think i figured a way to prove this. thanks for the help though. your solution seems more terse than mine. |

Re: continuous at any point iff continuous at originI'm flattered |

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