## Discontinuity and inequalities.

Limits, differentiation, related rates, integration, trig integrals, etc.
sepoto
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### Discontinuity and inequalities.

Question 1D-5:
Define $f(x)\, =\, \begin{cases}ax\, +\, b,&x\, \ge\, 1\\x^2&x\, <\, 1\end{cases}$
a) Find all values of $a$ and $b$ such that $f(x)$ is continuous.
b) Find all values of $a$ and $b$ such that $f'(x)$ is continuous. (Be careful!)

Solution 1D-5:
a) For continuity, we want $ax\, +\, b\, =\, 1$ when $x\, =\, 1$. Ans.: all $a,\, b$ such that $a\, +\, b\, =\, 1$.
b) $\frac{dy}{dx}\, =\, \frac{d(x^2)}{dx}\, =\, 2x\,=\, 2$ when $x\, =\, 1$. We have also $\frac{d(ax\, +\, b)}{dx}\, =\, a$. Therefore, to make $f'(x)$ continuous, we want $a\, =\, 2$. Combining this condition with the condition $a\, +\, b\, =\, 1$ from part (a), we get finally $b\, =\, -1,\, a\, =\, 2$.

I find myself lost on the concept that is being taught here. I'm thinking about and I see an inequality with what looks like a linear function and a parabolic function with a limit on the domain of each equation. The domain of the linear equation has to be greater than or equal to one and the domain of the parabolic has to be less than one.

The solutions manual seems to imply that the linear equation should be solved for all values of a and b where x = 1. My question is what about all the other values of the domain of the linear equation greater than one? I see that part b) involves the user of f'(x) but beyond that I don't really understand yet. I am familiar with f'(x) and take it to be the slope of the tangent line. Could I get some help in understanding this question a little better?

FWT
Posts: 153
Joined: Sat Feb 28, 2009 8:53 pm

### Re: Discontinuity and inequalities.

I find myself lost on the concept that is being taught here.
They're teaching that piecewise functions, to be continuous, have to match up at the breaks where the rules change.
The domain of the linear equation has to be greater than or equal to one and the domain of the parabolic has to be less than one.
It's not that it "has to be". It's just that it's defined that way; those are the rules on how this is set up.
My question is what about all the other values of the domain of the linear equation greater than one?
Polynomial functions are always continuous. The only place that this one might be discontinuous is where the rules change at x=1.