Editing Modulus of continuity (section)

===Elementary facts===
*If ''f'' has ω as modulus of continuity and ω<sub>1</sub> ≥ ω, then ''f'' admits ω<sub>1</sub> too as modulus of continuity.
*If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' are functions between metric spaces with moduli respectively ω<sub>1</sub> and ω<sub>2</sub> then the composition map <math>g\circ f:X\to Z</math> has modulus of continuity <math>\omega_2\circ\omega_1</math>.
*If ''f'' and ''g'' are functions from the metric space X to the Banach space ''Y'', with moduli respectively ω<sub>1</sub> and ω<sub>2</sub>, then any linear combination ''af''+''bg'' has modulus of continuity |''a''|ω<sub>1</sub>+|''b''|ω<sub>2</sub>. In particular, the set of all functions from ''X'' to ''Y'' that have ω as a modulus of continuity is a convex subset of the vector space ''C''(''X'', ''Y''), closed under [[pointwise convergence]].
*If ''f'' and ''g'' are bounded real-valued functions on the metric space ''X'', with moduli respectively ω<sub>1</sub> and ω<sub>2</sub>, then the pointwise product ''fg'' has modulus of continuity <math>\|g\|_\infty\omega_1+\|f\|_\infty \omega_2</math>.
*If <math>\{f_\lambda\}_{\lambda\in\Lambda}</math> is a family of real-valued functions on the metric space ''X'' with common modulus of continuity ω, then the inferior envelope <math>\inf_{\lambda\in\Lambda}f_\lambda</math>, respectively, the superior envelope <math>\sup_{\lambda\in\Lambda}f_\lambda</math>, is a real-valued function with modulus of continuity ω, provided it is finite valued at every point. If ω is real-valued, it is sufficient that the envelope be finite at one point of ''X'' at least.