I have a function, and I want to show that it is smooth. I know and are smooth. Does this imply that is smooth? If so, why?
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Originally Posted by Swlabr I have a function, and I want to show that it is smooth. I know and are smooth. Does this imply that is smooth? If so, why? No, not without further conditions, for instance g(x,y,z)=1, h(x,y,z)=1 are smooth, but f(x,y,z) is not even continuous. CB
Originally Posted by CaptainBlack No, not without further conditions, for instance g(x,y,z)=1, h(x,y,z)=1 are smooth, but f(x,y,z) is not even continuous. CB Good point. If I told you the function was would that help?...
Originally Posted by Swlabr Good point. If I told you the function was would that help?... That does not look continuous to me, put , and and consider the limit as . CB
Originally Posted by CaptainBlack That does not look continuous to me, put , and and consider the limit as . CB Forgetting the , we have . If we apply l'Hopital's rule we surely get that this tends to as . So that isn't a counter-example...
Originally Posted by Swlabr Forgetting the , we have . If we apply l'Hopital's rule we surely get that this tends to as . So that isn't a counter-example... Coming in along a ray where the limit is for any value of . (and how come you now have two 's?) L'Hopital's rule is not applicable if as the numerators are not zero. CB
Originally Posted by CaptainBlack Coming in along a ray where the limit is for any value of . (and how come you now have two 's?) L'Hopital's rule is not applicable if as the numerators are not zero. CB Hmm...I had forgotten about the conditions for l'Hopital's rule! I understand now. That is quite annoying! (my and are functions of which I couldn't be bothered writing out).
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