Advanced calculus

  • Thread starter mrs.malfoy
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  • #1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
97
0
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.
 
  • #3
336
0
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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