Advanced calculus

  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. 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. 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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