Definition: Modulus of Continuity of a Continuous Function

Let \((X,d_x)\) and \((Y,d_y)\) be metric spaces and let \(f:X\to Y\) be a continuous function on $X$. Using the $\epsilon-\delta$-definition of continuity, this means that for all $a\in X$ we have that for every $\epsilon > 0$ there is a $\delta > 0$ such that $d_y(f(\xi),f(a)) < \epsilon$ for all \(\xi\in X\) with $d_x(\xi,a) < \delta.$

The modulus of continuity of $f$, is the function $\omega_f:[0,\infty]\to[0,\infty]$, which assigns to each positive real number $\delta$ the supremum of distances $d_y(f(a),f(b))$ in $Y$ for all points $a,b\in X$, whose distance $d_x(a,b)$ is at most $\delta$ in $X,$ formally

$$\omega_f(\delta):=\sup\{d_y(f(a),f(b)):~a,b\in X,~d_x(a,b)\le\delta\}.$$

Roughly speaking, the modulus of continuity can be interpreted as the greatest positive number $\epsilon$ on whole $X$, for which for a given $\delta$ the $\epsilon-\delta$-property of continuity still holds for the function $f$. The concept was invented by Henri Lebesgue in 1910.

  1. Proposition: Modulus of Continuity is Subadditive
  2. Proposition: Modulus of Continuity is Monotonically Increasing
  3. Proposition: Modulus of Continuity is Continuous

Propositions: 1 2 3

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983