Questions tagged [derivatives-of-piecewise-functions]
8 questions
5
votes
3 answers
Can a function have a inflection point at somewhere non-differentiable?
Imagine this function :
$$
f(x)=\begin{cases}
x^2,\quad x<0 \\
\sqrt{x},\quad x\ge0.
\end{cases}
$$
In my lecture, my professor told me $f(x)$ is not differentiable at $0$ but $f(x)$ has an inflection point at $x=0$.
I get the idea that…
BRAD ZAP
- 425
- 6
3
votes
1 answer
Calculus: relationship between differentiability and continuity
I have a basic question. Consider the function:
$$f(x) = \begin{cases}
1-x^2 & \text{if $x\le 0$, and} \\
x^2 & \text{if $x > 0$.} \end{cases}$$
This function is not continuous at $0$: $\lim_{x\to0^{-}} f(x) = 1$, but $\lim_{x\to0^{+}} f(x) = 0$. So…
3
votes
2 answers
How to find the derivative of a piecewise function?
How to find the derivative of $\sqrt{x^{2}+4+3(x+\operatorname{sgn}(x))}$. That is find $\frac{\mathrm{d}}{\mathrm{d}x}(\sqrt{x^{2}+4+3(x+\operatorname{sgn}(x))})$. Now we clearly know that $\operatorname{sgn}(x)$ is a piecewise function.
We know…
user1203201
2
votes
1 answer
Finding Discounts for Multiple Items Accounting for Tax
I need a formula and/or a programmatic function (in any language) which will properly calculate a product discount so a customer can pay a price agreed upon at checkout, including tax, with multiple taxes applied to different amounts of the item…
Joe Davis
- 31
1
vote
2 answers
Differentiability of piecewise function at endpoint
Let $f$ be a function such that: $f: [\pi/4,\,\pi/2]\to \mathbb{R}$ $$ f(x)=\begin{cases} \frac{1}{\tan{x}}& \pi/4\leq x<\pi/2, \\ 0 & x=\pi/2. \end{cases} $$ How can I show that $f$ is differentiable on the closed interval $[\pi/4,\,\pi/2]$? I can…
No Name
- 31
1
vote
0 answers
Are these conditions on piecewise Lipschitz functions sufficient to make the overall function Lipschitz?
Let $K\subset \mathcal{R}^n$ be nonempty and closed. Let $f_1$ be locally Lipschitz on $\mathcal{R}^n$. Assume there is a finite $a=\inf_{x\in K^C}f_1(x)$. Suppose $f$ is defined as $f=a$ $\forall x\in K$ and $f=f_1(x)$ $\forall x \in K^C$. Also…
curiosity
- 453
1
vote
1 answer
Is this piecewise defined function locally Lipschitz?
Let $f:\mathcal{R}^n\rightarrow \mathcal{R}$ be continuous. Let $K\subset \mathcal{R}^n$ be nonempty and compact. Suppose $f$ is defined as $f=f_1(x)$ $\forall x\in K$ and $f=f_2(x)$ $\forall x\notin K$. Suppose $f_1$ and $f_2$ are locally Lipschitz…
curiosity
- 453
0
votes
2 answers
How to tell what is actually the derivative of a piecewise function
For example, if I have
$f(x)= \begin{cases} x & \text{if } x\neq0, \\ 0 & \text{if } x = 0. \end{cases}$
and I take the derivative:
$f'(x)= \begin{cases} 1 & \text{if } x\neq0, \\ 0 & \text{if } x = 0. \end{cases}$
then is the derivative of $f$ at…
Aqua Megami
- 21