Problem 436 (difficulty: 3/10)

The continuity of the function \(\displaystyle f:\R \to \R\) at the point \(\displaystyle a\) is defined by:

\(\displaystyle (\forall \varepsilon >0 ) (\exists \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) .\)

Consider the following variations of this formula.

\(\displaystyle (\forall \varepsilon >0 ) (\forall \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\exists \varepsilon >0 ) (\forall \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\exists \varepsilon >0 ) (\exists \delta >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\forall \delta >0 ) (\exists \varepsilon >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) ;\)

\(\displaystyle (\exists \delta >0 ) (\forall \varepsilon >0 )(\forall x)(|x-a|<\delta \akkor |f(x)-f(a )|<\varepsilon ) .\)

Which properties of \(\displaystyle f\) are described by these formulas?

Give me another random problem!