In a recent work, new functional spaces were introduced. These spaces characterize the fine local regularity of functions, and are defined through simple estimations on the pointwise values of the functions. In this work, we generalize their definition to the whole class of continuous functions, which allows us to reach every positive Hölder exponent. We prove that these new spaces coincide with the classical 2-microlocal spaces. It is interesting to notice that 2-microlocal spaces are originally defined in the frequency domain.. We prove the equality between both spaces in most of the cases. Using this result, we propose an algorithm able to estimate a part of the 2-microlocal frontier. From the 2-microlocal frontier can be extracted all the usual regularity exponents. Thus, as a by-product of the algorithm, robust estimators of, for instance, both the pointwise and the local exponents are obtained. Experiments on sampled data show that reasonnable accuracy is achieved even for «difficult» functions such as continuous but nowhere differentiable ones.