Example of non extensible functions ( $f(x) \notin EXT $) for reduction
Every total uncomputable function cannot be extended to a total computable function. More generally, let $g$ be an uncomputable function, and let $h$ be a total computable one-to-one function, for example $h(x) = 2x$. Define a partial function $f$ by $f(h(x)) = g(x)$ (so $f$ is defined only on the range of $h$). The function $f$ cannot be extended to a total computable function.