Issue with power fuse

Hi there, bit of an odd one. Perhaps someone with a bit more electronics background could advise what the issue is here.

Issue: PSP doesn't power on. Charges fine.

On investigating the issue I checked the power fuse (Marked TB) with a multimeter to see if that was the issue. The fuse showed continuity so assumed it was fine.

Then by accident I checked the fuse again while a battery was connected and as I test the fuse with the multimeter the PSP powers on.

Is the fuse the issue? Seems when I bridge it that it powers on automatically, but why if the fuse is giving me audible continuity when I check it?