Let f: [a,b] ---> R be continuous on [a,b] and differentiable on (a,b). Assume that there exists 0 ≤ M < + ∞ such that |f'(x) | ≤ M, for all x ∈ (a, b). Show that f is Lipschitz continuous on [a,b].

Let f: [a,b] ---> R be continuous on [a,b] and differentiable on (a,b). Assume that there exists 0 ≤ M < + ∞ such that |f'(x) | ≤ M, for all x ∈ (a, b). Show that f is Lipschitz continuous on [a,b].

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

The function f(x,y) x+y x²+y² define it at the origin such that the fis continuous at the origin? is not de_ned at the origin. Is it possible to
Please teach me how to prove f is continuous by using the definition below. Let f:R →R be defined by f(x)= 2x-1. Definition: A function f:R→R is continuous if for each open subset V of R, f^-1(V) is an open subset of R.
Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4 y y 3.0 2.5 2.0 1.5 1.0 0.5 y 3.0 2.5 2.0 1.5 1.0 0.5 W и 1 2 1 ^ 3 2 3 4 5 4 5 6 X X 2.5 W 2.0 1.5 3.0 1.0 0.5 y 3.0 2.5 2.0 1.5 1.0 0.5 1 T 2 3 2 3 4 5 4 5 сл 6 X X
Let f be a differentiable function defined on [-6, -2]. If f satisfies the conditions below, = -5 is the only critical number of f on [-6,-2). . "(-5) = -5 what can you conclude regarding f? MCQ13.pdf O The absolute maximum value of f on [-6,-2] is either f(-6) or f(-2). O The absolute minimum value of f on -6,-2] is f(-5). O The absolute maximum value of f on -6, -2 is also its local (or relative) maximum value. Of has no absolute minimum value on -6, -2.
Could explain how to do this in detail?
Let X and Y be two topological sPaces. Let A and B be such that Ā=B. let f:X » Y be a Continuous function. Show that F(A)=f (B).
Use the definition of continuity at a point to determine if f(x) = (x+2x')' is continuous at x =1. Need to satisfy three criteria!
Let f be continuous on [-1,5], where f(-1) = 12 and f(5) = 6. Can we conclude that a value -1 < c < 5 exists such that f(c) = 2?
Every function which is riemann integrable on [a, b] is continuous.true or false
Let f be a differentiable function. If f'(a) > 0 and f'(b) < 0 for some points a, b with a ≤ b, explain that there is some c in (a, b) with f'(c) = 0. (Note that f' might not be continuous, so the original Intermediate Value Theorem doesn't apply.)