We give and prove two new and fundamental properties of total-variation-minimizing function regularization (TV regularization): edge locations of function features tend to be preserved, and under certain conditions are preserved exactly; intensity change experienced by individual features is inversely proportional to the scale of each feature. We give and prove exact analytic solutions to the TV regularization problem for simple but important cases. These can also be used to better understand the effects of TV regularization for more general cases. Our results explain why and how TV-minimizing image restoration can remove noise while leaving relatively intact larger-scaled image features, and thus why TV image restoration is especially effective in restoring images with larger-scaled features. Although TV regularization is a global problem, our results show that the effects of TV regularization on individual image features are often quite local. Our results give us a better understanding of what types of images and what types of image degradation are most effectively improved by TV-minimizing image restoration schemes, and they potentially lead to more intelligently designed TV-minimizing restoration schemes.