Questions tagged [derivatives-of-piecewise-functions]
10 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 when $x<0$…
BRAD ZAP
- 425
4
votes
2 answers
Approximating piecewise smooth curves by smooth ones
I have been told that any piecewise smooth curve in the plane can be approximated by a sequence of smooth curves whose length converges to the one of the original curve. As long as I know it is done by removing the vertices and making them smooth.…
Joaquín Gavira López
- 71
- 4
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 $\lim_{x\to0^{+}} f(x) = 0$ .…
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…
curiosity
- 453
0
votes
0 answers
are smoothstep transition function and hermite splines equally in this smoothing problem?
I have trouble understanding the difference in the following problem. I have the function F(x) which is not differentiable at x=a $$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for $x<a$}\\ G(x) & \text{ for $x\geq a$}\end{array}\right.$$ and i…
Chris B.
- 153
- 6
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