This equality is not true. E.g., if $f(u)=u$, $x=0$, and $a=1$, then the left-hand side of the equality is $1$, whereas its right-hand sided is $2$.
The OP has now switched the meaning of $r$ from radius to diameter. The equality is still not true. E.g., if $f(u)=|u|$, $x=0$, and $a=1$, then the left-hand side of the equality is $1$, whereas its right-hand sided is $1/2$.