Continuous Function

by golb0016
Tags: continuous, function
golb0016 is offline
Nov30-09, 06:45 PM
P: 16
1. The problem statement, all variables and given/known data
find k for the function so it is continuous and differentiable.
x^2-1 x<=1
k(x-1) x>1

3. The attempt at a solution

k(x-1)=0 for x=1
k = 0/0?

How do I know if the function is differentiable?
Phys.Org News Partner Science news on
NASA's space station Robonaut finally getting legs
Free the seed: OSSI nurtures growing plants without patent barriers
Going nuts? Turkey looks to pistachios to heat new eco-city
statdad is offline
Nov30-09, 06:51 PM
HW Helper
P: 1,344
Work on making it differentiable at x = 1 first - if you can, then you know it will be continuous there. (what are the left / right-hand derivatives)
n!kofeyn is offline
Nov30-09, 09:45 PM
P: 538
Actually, that function is continuous for any value of k. The only point you have to worry about, both for continuity and differentiability, is the point x=1. You can explicitly show that the function is continuous at 1 by computing the limit of f as x approaches 1 and show that it equals f(1). You'll see it doesn't matter what k is for continuity.

To worry about differentiability, all you need to check is that the slope of the function k(x-1) at x=1 matches up with the slope of x2-1 at x=1. Remember that for a function to be differentiable at a point, the limit of the difference quotient at that point must exist. By checking that the slopes match up, you are checking that the left and right handed limits of the difference quotient equal each other (i.e., checking that the limit exists). The function is differentiable everywhere else since it is defined there by polynomials.

golb0016 is offline
Dec1-09, 10:45 AM
P: 16

Continuous Function

f(x) = x^2-1
f'(1) = 2(1) = 2

f(x) = k(x-1)
f'(1) = k = 2

Therefore k=2 and is differentiable at this point now, correct?

Register to reply

Related Discussions
Is the anti-derivative of a continuous function continuous? Calculus 13
continuous limited function, thus uniformly continuous Calculus & Beyond Homework 0
continuous function Set Theory, Logic, Probability, Statistics 2
Continuous function from Continuous functions to R Calculus & Beyond Homework 2
Continuous Function Introductory Physics Homework 8