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Advanced calculus 
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#1
Mar1510, 01:33 AM

P: 3

A function f is said to be symmetrically continuous at X_{0} if
lim [f(X_{0} + h)  f(X_{0}  h)]= 0 h> 0 Show that if f is continuous at X_{0}, it is symmetrically continuous there but not conversely. 


#2
Mar1510, 07:59 AM

P: 97

This sounds like homework so I'm not going to go into too much detail, but note that if f is continuous at x then: [tex]lim_{h\rightarrow0}f(x+h)=lim_{h\rightarrow0}f(xh)=f(x).[/tex]
There isn't much more to do. 


#3
Mar1710, 04:56 AM

P: 336

For the converse, take
f(x) =x if x is nonzero , f(0) =1. f is symmetrically continuous at 0, but not continuous. 


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