In a paper On the Gradient Inequality, S. Łojasiewicz and M.-A. Zurro give an example of a subanalytic function which is unbounded near zero, but continuous at this point on any analytic arc. However, their example turns out to be wrong, as we will show in this talk. Moreover, we prove that no such example exists in the subanalytic category, i.e. any subanalytic function that is arc-continuous at a point is continuous there. In addition, several new results concerning the Łojasiewicz exponent will be presented and discussed.