Interval of continuity and differentiability

• Aug 21st 2012, 09:15 AM
NFS1
Interval of continuity and differentiability
Stuck on a question which i do not know how to solve. Been trying to teach myself how do it but without any luck. Can some one please explain step by step how to solve the problem so i can try and solve similar problems.

Find the interval of continuity and differentiability of the following function:

f(x) = (\sqrt (x^2 - x)) + (\frac (1 / ln(2x - 1)))
• Aug 21st 2012, 10:17 PM
earboth
Re: Interval of continuity and differentiability
Quote:

Originally Posted by NFS1
Stuck on a question which i do not know how to solve. Been trying to teach myself how do it but without any luck. Can some one please explain step by step how to solve the problem so i can try and solve similar problems.

Find the interval of continuity and differentiability of the following function:

f(x) = (\sqrt (x^2 - x)) + (\frac (1 / ln(2x - 1)))

1. The square-root function and the ln-function are both continuous and differentiable in their respective domains. So determine the domains of these partial functions and afterwards determine the domain of f.

2. The square-root function is continuous and differentiable if

$\displaystyle x^2-x\ge 0~\implies~x\le0~\vee~x\ge1$ .......... [A]

3. The ln-function is continuous and differentiable if

$\displaystyle 2x-1>0~\implies~x\ge\frac12$.......... ["B"]

4. The denominator of the fraction must be unequal to zero: $\displaystyle \ln(2x-1)=0~\implies~2x-1=1~\implies~x=1$.......... [C]
That means: x = 1 doesn't belong to the domain of f.

5. Now "construct" a valid domain and write it in interval form.