<p>We discuss explicit approximations of uniformly continuous functions by Lipschitz continuous functions. Such constructions are relevant in the analysis of Toeplitz operators. As model cases, we treat <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {C}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with the Euclidean metric and the complex unit ball <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {B}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> equipped with the Bergman metric in details. In both cases a flow of approximating real analytic Lipschitz functions with a control of the Lipschitz constants is defined. We provide two applications of these results to semi-commutator estimates in Berezin Toeplitz quantization as well as to the analysis of certain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> algebras generated by Toeplitz operators on the complex unit ball <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\mathbb {B}}^2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">B</mi> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>