Definition: Uniformly Continuous Functions (General Metric Spaces Case)

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A function $f:X\mapsto Y$ is called uniformly continuous, if and only if for every $\epsilon > 0$ there is a $\delta > 0$ such that $$d_Y(f(x),f(y)) < \epsilon$$ for all $x,y\in X$ with $$d_X(x,y) < \delta.$$

  1. Theorem: Continuous Functions on Compact Domains are Uniformly Continuous

Proofs: 1
Theorems: 2


Thank you to the contributors under CC BY-SA 4.0!

Github:
bookofproofs


References

Bibliography

  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984