Let $f : [ 0,1 ] \longrightarrow [ 0,1 ]$ be a continuous function. Which of the following is/are true? (A) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) = 0$ and $g ( 1 ) = 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$. (B) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) < 0$ and $g ( 1 ) > 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$. (C) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $0 < g ( 0 ) < 1$ and $0 < g ( 1 ) < 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$. (D) For every continuous $g : [ 0,1 ] \longrightarrow [ 0,1 ]$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.
Let $f : [ 0,1 ] \longrightarrow [ 0,1 ]$ be a continuous function. Which of the following is/are true?\\
(A) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) = 0$ and $g ( 1 ) = 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.\\
(B) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $g ( 0 ) < 0$ and $g ( 1 ) > 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.\\
(C) For every continuous $g : [ 0,1 ] \longrightarrow \mathbb { R }$ with $0 < g ( 0 ) < 1$ and $0 < g ( 1 ) < 1$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.\\
(D) For every continuous $g : [ 0,1 ] \longrightarrow [ 0,1 ]$ there exists $x \in [ 0,1 ]$ with $f ( x ) = g ( x )$.