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 + Δu) − F(u)) / Δu exists as Δu → 0. For a vector function, this simply means that each component has a derivative. The derivative is then a new vector made up of those component derivatives.

4) Key facts

• If a function is differentiable at a point, it is automatically continuous there — but not the other way around.
• Basic limit laws apply to each component: sums and products of continuous functions remain continuous, and the same logic extends to derivatives.
• Any norm on ℝ³ gives the same definition of continuity, since all norms are equivalent in finite-dimensional spaces.

5) Quick checks

• Continuity check: Do the component limits equal their corresponding values at u?
• Differentiability check: Do the component difference quotients approach finite limits?
• Note: A curve may be continuous but not differentiable at sharp corners or cusps.

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Date: 01.11.2025