I study the definition of upper and lower derivative in the book Real Analysis H.L. Royden 4th edition.
$\overline{D}f(x)=\lim\limits_{h \to 0}\left[\sup\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$ is definition of upper derivative
$\underline{D}f(x)=\lim\limits_{h \to 0}\left[\inf\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$ is definition of lower derivative
For me, this definition feels nonsense because when h is negative $0<|t|\leq h$ part looks undefined. So, I think correct definition of upper derivative is $\overline{D}f(x)=\lim\limits_{h \to 0+}\left[\sup\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\right]$
I think I'm missing very trivial point but I cannot figure out that point myself.
I would appreciate your reply.