These  Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson

A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

Proofs of  one-sided theorems could be of interest in the study of 2D topology.

Velocity Revisited

In the discussion of distance versus time, suppose d = f(t). Further suppose v_min £ v = f¢(t) = velocity £ v_max for t₁ £ t £ t₂ = t₁+Dt. Then the mean value theorem implies there is a time t = c such that the average speed

f(t2)-f(t1) t2-t1 = f¢(c)

Therefore [(f(t₂)-f(t₁))/(t₂-t₁)] ³ v_min. Thus the distance traveled

Dd = f(t2)-f(t1) ³ vmin·(t2-t1) = vmin·Dt

Similarly [(f(t₂)-f(t₁))/(t₂-t₁)] £ v_max and the distance traveled is

Dd = f(t2)-f(t1) £ vmax·(t2-t1) = vmax·Dt

What happens when v = f¢(t) is constant?

Finally observe,

vmin·(t2-t1) £ f(t2)-f(t1) £ vmax·(t2-t1)

and K = max(|v_min|, |v_max|) implies that

|f(t2)-f(t1)| £ K (t2-t1)

whenever t₁ < t₂.

The foregoing indicates d = f(t) is Lipschitz continuous on the time-interval a £ t £ b if the velocity v = f¢(t) with magnitude |f¢(t)| £ K < ¥ on the time interval [a,b]. Thus particle motions with bounded velocity are described by Lipschitz continuous functions.

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