sum += (x > 0 && y > 0 && x < x_limit && y < y_limit) ? a[y][x] : 0;

That took less than a minute of thought; or you could use a common technique in image and video processing, which is to make sure that the edges are always present but filled with appropriate values by defining your grid to be slightly bigger than the active area.

And it doesn’t do what’s asked? Congrats.

I just concentrated on what would be the hardest part, the "edge" cases (literally.) Wrapping that in a loop that goes through the appropriate indices is not much more work. There is a sibling dead comment to mine which has the rest.

Should probably use

  >=

You're assuming closed instead of open intervals.

I was assuming zero-based arrays.

oops doesn't count the edges. Take another minute.

-- joking -- :)