Calculate the normal cone of a convex set at a point Let $C$ be a convex set in $\mathbb{R}^d$ and $\overline{x}\in C$. We define the normal cone of $C$ at $\overline{x}$ by 
\begin{equation}
 N_C(\overline{x}) = \{ y \in \mathbb{R}^d \ <y ,c-\overline{x}> \leq 0 \forall c \in C \}.
\end{equation}
I found in a book of nonsmooth analysis that using this definition the normal cone of 
\begin{equation}
 C= \{ (x,0) \in \mathbb{R}^2 : 0\leq x \leq 1 \} 
\end{equation}
at $\overline{x} = (0,0)$  is the set
\begin{equation}
 N_C(0,0) = \{ (y_1,y_1) \in \mathbb{R}^2 : y_1\leq 0 , y_2 \in \mathbb{R} \}.
\end{equation}
My question is: How to calculate the normal cone? I tried to use the definition, but I couldn't obtain the result!
Thank you very much.

In your case
\begin{align}
N_C(0,0) &= \{ (y_1,y_2)\in \mathbb{R}^2|y_1c_1+y_2c_2\le 0, \forall (c_1,c_2)\in C\}\\
&=\{ (y_1,y_2)\in \mathbb{R}^2|y_1c_1\le 0, \forall\, c_1\in [0,1]\}\\
&= \{ (y_1,y_2)\in \mathbb{R}^2 | y_1\le 0\}
\end{align}
