Continuity and Differentiability (Clear, Compact Guide)
Continuity and Differentiability (Clear, Compact Guide) We study functions of a single real parameter u . A scalar function is written as φ(u) , and a vector function as R(u) = (x(u), y(u), z(u)) . All the ideas below are based on ordinary one-variable limits. 1) What it means for a function to be continuous at u (scalar case) A function φ is said to be continuous at u if its value changes smoothly as u changes slightly. Formally, for every ε > 0 , there exists a δ > 0 such that |φ(u + Δu) − φ(u)| < ε whenever |Δu| < δ . 2) Continuity of a vector function Let R(u) = (x(u), y(u), z(u)) . The function R is continuous at u if each component x(u) , y(u) , and z(u) is continuous at that same point. Equivalently, using any fixed norm |·| on ℝ³ : for every ε > 0 , there exists a δ > 0 such that |R(u + Δu) − R(u)| < ε whenever |Δu| < δ . 3) Differentiability (first order) A scalar or vector function is differentiable at u if the limit (F(u + Δ...