Piece-Wise Function Continuity

Click For Summary
SUMMARY

The discussion focuses on determining the continuity of the piece-wise function defined as f(x) = {x^3 if x ≤ -2, 2 if x > -2}. The function is continuous at x = -2 if the three conditions for continuity are satisfied: f(c) exists, the limit as x approaches c exists, and the limit equals f(c). The left-hand limit as x approaches -2 is -8, while the right-hand limit is 2, indicating that the function is discontinuous at x = -2 due to the failure of the limit condition.

PREREQUISITES
  • Understanding of piece-wise functions
  • Knowledge of limits in calculus
  • Familiarity with continuity conditions in mathematics
  • Basic algebra skills for evaluating functions
NEXT STEPS
  • Study the concept of limits and how to calculate them for piece-wise functions
  • Learn about the formal definition of continuity in calculus
  • Explore examples of discontinuous functions and their classifications
  • Practice evaluating limits from both sides for various functions
USEFUL FOR

Students studying calculus, particularly those learning about continuity and limits, as well as educators looking for examples of piece-wise function analysis.

justapinkday
Messages
1
Reaction score
0

Homework Statement


Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold.

f(x)= {x^3 if x < or = -2
{2 if x > -2


Homework Equations


The conditions are that a function is said to be continuous at x=c if the following conditions hold:
1 f(c) exists
2 lim as x approaches c f(x) exists
3 lim as x approaches c f(x) = f(c)



The Attempt at a Solution


My professor has not explained this and has assigned it...no clue what to do!
 
Physics news on Phys.org
You will have to calculate the limit in each point and see whether the left and right sided limits are equal.
 

Similar threads

Use Graph To Investigate Limit
  • · Replies 7 ·
Replies
7
Views
1K
Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
  • · Replies 20 ·
Replies
20
Views
3K
Medium Hard continuity proof Tutorial Q6
  • · Replies 2 ·
Replies
2
Views
1K
A few questions to make sure I understand Fourier series
  • · Replies 3 ·
Replies
3
Views
2K
Proving piecewise function is k-differentiable
  • · Replies 5 ·
Replies
5
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Show that the given function is continuous
  • · Replies 27 ·
Replies
27
Views
2K
f(x) = 2x+1, proving that it is continuous when p = 1 with 𝛿 and ε
  • · Replies 7 ·
Replies
7
Views
2K
Prove that Lim x->c f(x)g(x)=0 if Lim x->c f(x)=0 and |g(x)|<K
  • · Replies 2 ·
Replies
2
Views
2K
How to show a function is twice continuously differentiable?
  • · Replies 3 ·
Replies
3
Views
2K