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Advanced calculus

by mrs.malfoy
Tags: advanced, calculus
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Mar15-10, 01:33 AM
P: 3
A function f is said to be symmetrically continuous at X0 if

lim [f(X0 + h) - f(X0 - h)]= 0
h-> 0

Show that if f is continuous at X0, it is symmetrically continuous there but not conversely.
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Mar15-10, 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(x-h)=f(x).[/tex]

There isn't much more to do.
Mar17-10, 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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