ok  firstly try drawing rough graph

lim x tend to 0..

RHL = [ h sinh ] = 0

LHL = [ -h sin( - h) ] = 0

so it is continuous at 0.

by inspection we see that it is continuous in (-1,1)

check at x = 1

LHL = [ (1 + h) sin(h + 1) ] = [-(1 + h) sinh] =-1

RHL = [(1 -h) sin(1 -h)]  =  [(1 - h) sinh ] = 0 

so it is discontinuos at x = 1