Given a piecewise, prove that it is continous and differentiable

  • Thread starter Thread starter NWeid1
  • Start date Start date
  • Tags Tags
    Differentiable
Click For Summary
SUMMARY

The function f(x) defined as f(x) = sin(x)/x for x≠0 and f(0) = 1 is continuous and differentiable for all x. To prove continuity at x=0, the limit of f(x) as x approaches 0 must equal f(0), which is satisfied since lim (x→0) sin(x)/x = 1. Furthermore, the derivative f'(x) exists for all x, including at x=0, confirming that f'(x) is also continuous across its domain.

PREREQUISITES
  • Understanding of limits and continuity in calculus
  • Knowledge of differentiability and the definition of the derivative
  • Familiarity with piecewise functions
  • Basic trigonometric functions and their properties
NEXT STEPS
  • Study the formal definition of continuity and differentiability in calculus
  • Learn how to evaluate limits involving trigonometric functions
  • Explore the concept of piecewise functions and their derivatives
  • Investigate the implications of differentiability on continuity
USEFUL FOR

Students studying calculus, particularly those focusing on continuity and differentiability of functions, as well as educators looking for examples of piecewise function analysis.

NWeid1
Messages
81
Reaction score
0
1. Homework Statement
For f(x)= { sin(x)/x if x≠0 , 1 if x=0. (a) Show that f is continuous and differentiable for all x. (b) Show the derivative f'(x) is continous.


2. Homework Equations



3. The Attempt at a Solution
I know that if f is differentiable it is continous, so I need to focus on x=0 to show that the function is differentiable. And now I'm stuck lol
 
Physics news on Phys.org
Write down the definition of the derivative at x=0.
 

Similar threads

Proving piecewise function is k-differentiable
  • · Replies 5 ·
Replies
5
Views
2K
Show that the given function is continuous
  • · Replies 27 ·
Replies
27
Views
2K
Non-Differentiable Function proof
  • · Replies 8 ·
Replies
8
Views
1K
Show function series involving arctan is not differentiable at x=0
  • · Replies 26 ·
Replies
26
Views
3K
Discontinuous partial derivatives example
  • · Replies 2 ·
Replies
2
Views
2K
Given a piecewise, prove that it is continous and differentiable
  • · Replies 1 ·
Replies
1
Views
1K
Given unit impulse response, obtain differential equation
  • · Replies 7 ·
Replies
7
Views
2K
Is this the correct general solution of the given PDE?
  • · Replies 12 ·
Replies
12
Views
2K
Linearly independent functions with identically zero Wronskian
  • · Replies 1 ·
Replies
1
Views
2K
Show that this graph isn't differentiable at x=1
  • · Replies 22 ·
Replies
22
Views
3K