The following is inside my GSXR600, but I'm pretty sure it's very similar to what's in the SV.
This is a shift fork:
It slides these gears left and right so you can shift:
If one or more of these forks are bent, the gears won't slide far enough and won't mesh with the other gears properly, so you can't shift.
Or you might have the same problem I did. I'm not sure though, because I couldn't shift out of 4th gear. The shifter would just move up and down, but no shifting. Look at the pretty pictures:
The circled bolt was loose, causing the shift star (the funny shaped cam with the posts sticking out of it) to just wobble back and forth instead of turning the shift cam (the slotted barrel that moves the forks into position for each gear). Once I loctited and tightened the bolt, it looked like this:
This concludes today's session of "Here's what your problem might be and why it's an issue."
Good luck.