If X is a Asplund space and $f:X\rightarrow \bar{\mathbb{R}}$, $C=\{x:f(x)\leq 0\}, \bar{x}\in C?_help_Articles

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If X is a Asplund space and $f:X\rightarrow \bar{\mathbb{R}}$, $C=\{x:f(x)\leq 0\}, \bar{x}\in C?

If X is a Asplund space and f:X→R, C={x:f(x)≤0},x¯∈C,f(x¯)=0C={x:f(x)≤0},x¯∈C,f(x¯)=0, is limiting subdiffere

If X is a Asplund space and f:X→R, C={x:f(x)≤0},x¯∈C,f(x¯)=0C={x:f(x)≤0},x¯∈C,f(x¯)=0, is limiting subdifferential ff at x¯x¯ subset of limiting normal cone to CC at x¯x¯? i.e. ∂Lf(x¯)⊆NL(x¯,C)?∂Lf(x¯)⊆NL(x¯,C)? with which conditions it holds?

or:If X is a Asplund space and f:X\u2192R, C={x:f(x)\u22640},x\u00af\u2208C,f(x\u00af)=0C={x:f(x)\u22640},x\u00af\u2208C,f(x\u00af)=0, is limiting subdifferential ff at x\u00afx\u00af subset of limiting normal cone to CC at x\u00afx\u00af? i.e. \u2202Lf(x\u00af)\u2286NL(x\u00af,C)?\u2202Lf(x\u00af)\u2286NL(x\u00af,C)? with which conditions it holds?

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