\[f(x)=-f(-x)\]
\[f(x)=f(-x)\]
respectively.
When you differentiate these equations you get
\[f'(x)=--f'(-x)=f'(-x)\]
(1)\[f'(x)=-f'(-x)\]
(2)Differentiating an odd function gives an even function and vice versa. The same is not true when integrating in general, because of the role of the arbitrary constant .Integrating(1)and(2)gives
\[f(x)=-f(-x)+c\]
\[f(x)=f(-x)+c\]
so the odd and even conditions fail without the further condition that
\[c=0\]
.