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.

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: $$lim_{h\rightarrow0}f(x+h)=lim_{h\rightarrow0}f(x-h)=f(x).$$

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.