While writing a story which happens to be set on Pan, one of Saturn's moons, I realised that I had made an erroneous statement in a past blog post. Of course, I had to correct it.
In that post I made a passing comment that on Pan, the gravitational force of Saturn is greater than that of the moon itself. That statement was true. However, I went on to say that on the Saturn-side of Pan, you'd fall off because the gravity of Pan wasn't strong enough to overcome Saturn's gravity. This last part isn't quite true.
If you were on Pan, despite its weak gravity, you would be hurtling around Saturn at the same speed that Pan does, which means that you would automatically be going fast enough to stay in orbit (your centripetal acceleration would be balancing Saturn's acceleration due to gravity), no matter which side of Pan you were on. Admittedly, the low escape velocity (about 25 km/h) and the slight difference in gravitational pull from Saturn between the near and far ends would make it easy to fall off the planet and slowly spiral in towards Saturn, but you certainly wouldn't be falling upwards.
The more accurate statement I should've made was that if you were floating around in the vicinity of Pan's orbit and Pan came past you, its gravity would not be strong enough to pull you in over Saturn's gravity. No matter how close to it you were (even if you could touch the surface), if you were not already moving along with it, then Saturn's gravity would win out and you would fall towards Saturn, rather than towards Pan.
The original post has been amended to reflect the above correction.