Final answer:
The false statement could potentially be that the function f is continuous on the entire real number line. Differentiability at a point ensures local continuity and the existence of a derivative at that point, but not globally on the real line nor at every point in its domain.
Step-by-step explanation:
If a function f is differentiable at a point x=a, the correct statement that could potentially be false is: d) The function f is continuous on the entire real number line.
Being differentiable at a point means the function has a derivative exactly at that point and is continuous at that point as well. This is due to the fact that differentiability at a point implies continuity at that point (a), but does not necessarily mean the function is continuous everywhere else on the real line (d), nor that it has a derivative at every other point in its domain (c). Also, differentiability at a point does not provide information about whether the function has a local minimum there (e).
The function can be differentiable at a point while still having discontinuities elsewhere, only guaranteeing local behavior around x=a. It could potentially have a discontinuity at any other point on the real number line, or it could simply not be defined everywhere. Thus, d) is the statement that could potentially be false.