Equicontinuity

A family $\mathcal{F}$ of maps between metric spaces is equicontinuous when, near each point, one input tolerance controls the output variation uniformly for every $f\in \mathcal{F}$. Uniform equicontinuity uses one input tolerance over the whole domain.

A family $\mathcal{F}$ of maps between metric spaces is equicontinuous at $x$ when, for every $\epsilon >0$, one $\delta >0$ works for all $f\in \mathcal{F}$: $d(x,y)<\delta$ implies $d(f(x),f(y))<\epsilon$.

A family $\mathcal{F}$ of scalar-valued functions is uniformly bounded when one constant $M$ satisfies $|f(x)|\le M$ for every $f\in \mathcal{F}$ and every point $x$ in their common domain.

A family $\mathcal{F}$ of scalar-valued functions on $X$ is pointwise bounded when ${\sup }_{f\in \mathcal{F}}|f(x)|<\infty$ for every fixed $x\in X$; the bound may depend on $x$.

For a compact Hausdorff space $K$, a subset of $C(K)$ is relatively compact in the uniform norm exactly when it is equicontinuous and pointwise bounded. On a compact metric space, this says that every uniformly bounded equicontinuous sequence has a uniformly convergent subsequence.

If a thrice differentiable solution has both endpoint values and both endpoint first derivatives bounded in modulus by $N$, it obeys $|f|\le N$ and $|f^{\prime }|\le N$ throughout the interval. A positive interior maximum of $f$ would give $f^{\prime \prime }=f>0$, and a negative interior minimum would give $f^{\prime \prime }=f<0$, both impossible. For $w=f^{\prime }$, differentiation gives $w^{\prime \prime }=w+2w{w}^{\prime }$, so the same argument applies at an extremum of $w$. The resulting uniform bound and Lipschitz constant imply total boundedness of the solution family by the Arzelà-Ascoli theorem.

A subset of a topological space is relatively compact when its closure is compact.

If successive subsequences converge uniformly on an exhaustion $K_1\subset K_2\subset \cdots$, the diagonal sequence converges uniformly on every fixed $K_m$, hence locally uniformly on their union.

For a compactly supported nonzero function $\psi$, the translates $f_n(x)=\psi (x-n)$ converge uniformly to zero on every compact subset of $\mathbb{R}$, while $\Vert f_n{\Vert }_{\infty }=\Vert \psi {\Vert }_{\infty }$ prevents uniform convergence on all of $\mathbb{R}$.