Suppose $f(x,y)$ is a total computable function. For each $m$, let $g_m$ be the computable function given by $g_m(y) = f(m,y)$.
Construct a total computable function h such that for each $m$, $h \not= g_m$.
My work:
Is this a really simple question where I can just use the diagonal method?
Define
$ h(m,y) = \begin{cases} g_m(y)+1 & \mbox{if $g_m(y)$ is defined} \\ 0 & \mbox{ if $g_m(y)$ is undefined} \end{cases} $
Then clearly, $h$ differs from $g_m$ at all points. I'm just not entirely sure if $h$ as I constructed it is total and computable.