Show that $H_{N}$ is positive, continuous on $\Sigma_{N}$ and calculate the value of $H_{N}(p)$ when $p_{i} = 1/N$ for all $i \in \{1, \ldots, N\}$ (uniform distribution on $\{1, \ldots, N\}$).
Where $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$ and $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$