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