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Advanced calculus

  1. Mar 15, 2010 #1
    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.
     
  2. jcsd
  3. Mar 15, 2010 #2
    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.
     
  4. Mar 17, 2010 #3
    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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