Writing the angular boundary limits (non-tangential boundary limits) of inner and outer functions in the unit disc in the form f(e it) = ρ(t)e iθ(t) , 0 ≤ t ≤ 2π, the paper studies the sign-change property of the “phase derivative” that reduces to θ 0 (t) if the function has an appropriate parameterization in t. We show in the inner functions case the Julia-Wolff-Carath´eodory Theorem may be rephrased to conclude the positivity property of the phase derivative. On the other hand, outer functions do not have such property. In the introduction we indicate that this study is motivated by the concept instantaneous frequency and relevant study in contemporary signal analysis.