**4. Fuzzy function**

In this section, let us define, in general, a fuzzy function of one real variable as follows:

$$
\tilde{\psi}: \mathcal{D} \to \mathcal{F}(\mathbb{R}): \mathfrak{x} \mapsto \tilde{\psi}(\mathfrak{x}) \tag{22}
$$

where Fð Þ the set of fuzzy functions defined on .

a. *Alpha-cut of ψ*~ð Þ *x*

The *α*-cup representation of *ψ*~ð Þ *x* is:

$$\tilde{\boldsymbol{\varphi}}(\boldsymbol{\pi})\_a = \tilde{\boldsymbol{\varphi}}(\boldsymbol{\pi}, a) = \left[ \tilde{\boldsymbol{\varphi}}^L(\boldsymbol{\pi}, a), \tilde{\boldsymbol{\varphi}}^U(\boldsymbol{\pi}, a) \right], a \in [0, 1] \tag{23}$$

b. *Kernel of ψ*~ð Þ *x*

The kernel of *ψ*~ð Þ *x* , also called modal of *ψ*~ð Þ *x* is defined by:

$$\ker\left(\check{\wp}\left(\boldsymbol{x}\right)\_{a=1}\right) = \check{\wp}^{L}\left(\boldsymbol{x},\mathbf{1}\right) = \check{\wp}^{U}\left(\boldsymbol{x},\mathbf{1}\right) \tag{24}$$

c. *Support of ψ*~ð Þ *x*

The support of *ψ*~ð Þ *x* is defined by:

$$\text{supp}\left(\ddot{\boldsymbol{\upmu}}(\mathbf{x})\_{\boldsymbol{a}=\mathbf{0}}\right) = \ddot{\boldsymbol{\upmu}}(\mathbf{x}, \mathbf{0}) = \left[\ddot{\boldsymbol{\upmu}}^{L}(\mathbf{x}, \mathbf{0}), \ddot{\boldsymbol{\upmu}}^{U}(\mathbf{x}, \mathbf{0})\right] \tag{25}$$
