upper semi-continuous of fuzzy set

Question

Let $u:\mathbb{R^n}\to [0,1]$ be a fuzzy set. (fuzzy set is a set of ordered pairs $(x,u(x)), x\in \mathbb{R^n})$. Please give an example such that $u(x)$ be upper semi-continuous. thanks

Answer 1

Let $\mathbf{x} = \{x_i\}^n_{i=1}$. The indicator function of the unit ball defined as $$u(\mathbf{x}) = \begin{cases} 1 & \text{if $\Vert \mathbf{x} \Vert_2 = \left( \sum^{n}_{i=1} x_i^2 \right)^{1\over2} \leqslant 1$} \\ 0 & \text{otherwise} \end{cases}$$ is upper semi-continuous. Another example would be $$v(\mathbf{x}) = \begin{cases} 1 & \text{if $0 \leqslant x_i \leqslant 1$ for all $i$}\\ 0 & \text{otherwise} \end{cases}$$ which would be the indicator function of a unit (hyper-)cube, also upper semi-continuous (thanks to the $\leqslant$ in the expression)

Note: this makes $(\mathbf{x},u(\mathbf{x}))$ and $(\mathbf{x},v(\mathbf{x}))$ examples of "crisp" fuzzy sets. They map $\mathbb{R}^n\to \{0,1\}$ which is a special case of $\mathbb{R}^n\to [0,1]$. If you need something with "more fuzziness", I can offer $$w(\mathbf{x}) = \begin{cases} \frac{1}{1+(\Vert \mathbf{x} \Vert_2)^2} & \text{if $\Vert \mathbf{x} \Vert_2 \leqslant 1$} \\ \frac{1}{1+4(\Vert \mathbf{x} \Vert_2)^2} & \text{otherwise} \end{cases}$$