Let us define a new function $g:$
$g(y) = f(y) -f(y+1)$
Since, function $f$ is continuous in $[0,2],$ $g$ would be continuous in $[0,1]$.
$g(0) = -2, g(1) = 2$
Since, $g$ is continuous and goes from negative to positive value in $[0,1],$ at some point $g$ would be $0$ in $(0,1).$
$g=0 \implies f(y) = f(y+1)$ for some $y \in (0,1).$
Therefore, correct answer would be $(A).$