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逐點收斂 ¨ç¼Æ¯Å¼Æ ¨ç¼Æ§Ç¦C ³vÂI¦¬ÀÄ §¡¤Ã¦¬ÀÄ §¡¤Ã¦¬ÀĪº©Ê½è ¨ç¼Æ¯Å¼Æ ¨ç¼Æ¯Å¼Æ ¨ç¼Æ§Ç¦C ²Å¸¹:³]{fn}¬°¤@©w¸q¦b¦³¬É°Ï¶¡(a,b)¤Wªº¹ê¼Æ¨ç¼Æ§Ç¦C. ³vÂI¦¬ÀÄ(pointwiseconvergence) ¦pªG¹ï¥ô·Nx(a,b),limnfn(x)¦s¦b,«hºÙfn(x)¦b(a,b)¤W³vÂI¦¬ÀÄ. ¨ä·¥­­¤]¬O¨ç¼Æ,¼g¬°f(x)=limnfn(x). ©w¸q. ¹ï¥ô·N>0,x(a,b),³£¬Û¹ï¦s¦b¥¿¾ã¼Æn0(x),¨Ï±o¹ï©Ò¦³n>n0(x),|fn(x)-f(x)|n,fn(x)=1;§_«h¨ä­È¬°0.«hf(x)0. f(x)¦b[0,)¤W³sÄò,»Pfn¤£¦P. f(x)¦b[0,)¤W¿n¤À¬°0,¦Ófn¦b[0,)¤W¤£¥i¿n¤À. ³]x[0,1],fn(x)=xn(¹Ï¥Ü). «h¹ïx[0,1),f(x)=0;f(1)=1. f(x)¦b[0,1]¤W¤£³sÄò,»Pfn¤£¦P. §¡¤Ã¦¬ÀÄ(uniformconvergence) ¦pªGfn(x)¦b(a,b)¤W³vÂI¦¬ÀÄ,¥B¦b¦¬ÀÄ©w¸q¤¤,n0»PxµLÃö,«hºÙfn(x)¦b(a,b)¤W§¡¤Ã¦¬ÀÄ. ©w¸q. ¹ï¥ô·N>0,³£¬Û¹ï¦s¦b¥¿¾ã¼Æn0,¨Ï±o¹ï©Ò¦³n>n0,x(a,b),|fn(x)-f(x)|