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Deep neural networks (DNNs) show promise in breast cancer screening, but their robustness to input perturbations must be better understood before they can be clinically implemented. There exists extensive literature on this subject in the context of natural images that can potentially be built upon. However, it cannot be assumed that conclusions about robustness will transfer from natural images to mammogram images, due to significant differences between the two image modalities. In order to determine whether conclusions will transfer, we measure the sensitivity of a radiologist-level screening mammogram image classifier to four commonly studied input perturbations that natural image classifiers are sensitive to. We find that mammogram image classifiers are also sensitive to these perturbations, which suggests that we can build on the existing literature. We also perform a detailed analysis on the effects of low-pass filtering, and find that it degrades the visibility of clinically meaningful features called microcalcifications. Since low-pass filtering removes semantically meaningful information that is predictive of breast cancer, we argue that it is undesirable for mammogram image classifiers to be invariant to it. This is in contrast to natural images, where we do not want DNNs to be sensitive to low-pass filtering due to its tendency to remove information that is human-incomprehensible.