Reply to Re: I gave up to the Computers National Olimpiad
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My trick for 4 was actually integrating it:
G(x) = G(ax) + G(bx), where G(x) = \int g(x) dx
Still need the continuity of g here. I believe Riemann-integrals of (mostly everywhere) continuous functions are continuous also, because its derivative exists and is equal to the original function.
Other than the way we came there, it's the same situation
Btw, not all singular matrices do the job, for example:
((1,0),(0,0)) * ((1,0),(0,0)) = ((1,0),(0,0))
((0,0),(0,1)) * ((0,0),(0,1)) = ((0,0),(0,1))
These matrices serve as their own identity.
As for the third one... more analysis, but it seems an awful lot like the limit definition, which is needed for continuity. I didn't really look at it yet.
G(x) = G(ax) + G(bx), where G(x) = \int g(x) dx
Still need the continuity of g here. I believe Riemann-integrals of (mostly everywhere) continuous functions are continuous also, because its derivative exists and is equal to the original function.
Other than the way we came there, it's the same situation
Btw, not all singular matrices do the job, for example:
((1,0),(0,0)) * ((1,0),(0,0)) = ((1,0),(0,0))
((0,0),(0,1)) * ((0,0),(0,1)) = ((0,0),(0,1))
These matrices serve as their own identity.
As for the third one... more analysis, but it seems an awful lot like the limit definition, which is needed for continuity. I didn't really look at it yet.
