Intereting Posts

Limit for entropy of prime powers defined by multiplicative arithmetic function
How many circles are needed to cover a rectangle?
Area under parabola using geometry
How to compute the gcd of $x+a$ and $x+b$, where $a\neq b$?
Local solutions of a Diophantine equation
Open Cover for a Compact Subset
Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$
Morley rank (with an unusual definition)
the knot surgery – from a $6^3_2$ knot to a $3_1$ trefoil knot
A functional relation which is satisfied by $\cos x$ and $\sin x$
Solving $x^2 \equiv 1 \pmod{p^{\ell}}$
How to write well in analysis (calculus)?
Express $\int\exp\left(\frac{a}{x}+bx\right)x^{\eta}\mathrm{d}x$ in terms of special functions?
Prove an interesting property of pdf moments?
Extend isometry on some cube vertices to the entire cube

Determine whether the following functions are differentiable, continuous, and whether its partial derivatives exists at point $(0,0)$:

(a) $$f(x, y) = \sin x \sin(x + y) \sin(x − y)$$

(b)$$f(x,y)=\sqrt{|xy|}$$

- The square of minimum area with three vertices on a parabola
- Convergence of a compound sequence
- Evaluating $\lim\limits_{x\to \infty}\sqrt{x^{6}+x^{5}}-\sqrt{x^{6}-x^{5}}$
- Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent
- Volume of a hypersphere
- Proof that $t-1-\log t \geq 0$ for $t > 0$

(c)$$f(x, y) = 1 − \sin\sqrt{x^2 + y^2}$$

(d) $$f(x,y) = \begin{cases} \dfrac{xy}{x^2+y^2} & \text{if $x^2+y^2>0 $} \\ 0 & \text{if $x=y=0$} \end{cases}$$

(e) $$f(x,y) = \begin{cases} 1 & \text{if $x y \ne 0$} \\ 0

& \text{if $xy=0$} \end{cases}$$

(f)$$f(x,y) = \begin{cases} \dfrac{x^2-y^2}{x^2+y^2} & \text{if $x^2+y^2>0$} \\ 0 & \text{if $x=y=0$} \end{cases}$$

My try:

For (a), using the definition of the derivative for a multivariate function, the limit tends to $0$, hence it’s differentiable and its partial derivative exists and it’s continuous.

For (b) I mentioned that is not differentiable as using the definition of the derivative for a multivariate function, the limit does not tend to $0$. While it’s continuous, as the limit of the function $f(x,y)$ tends to $0$. To determine whether its partial derivative exists, this part is tricky because of the modulus sign in the function hence I’m unsure whether a modulus is differentiable for this case.

For part (c) it should be continuous but not differentiable at $(0,0)$ because its partial derivative does not exists at $(0,0)$. As to why its partial derivatives does not exists, lets say to find $$f_x$$, we let $y=0$, the expression $f(x,y)$ becomes $1-\sin(|x|)$ which is not differentable at $(0,0)$, hence partial derivatives cannot exists at $(0,0)$.

For (d), this question is also tricky, as although initially I though its partial derivatives exists, now I think likewise. Because if I want to differentiate the function with respect to $x$ for example, I would sub in the value of $y=0$ making the numerator a zero and hence assume the derivative is $0$. However, on closer look, there is still the denominator of $x^2$ and if $x^2=0$ the denominator becomes $0$ and since $0/0$ is undefined, the partial derivatives do not exist. As for continuity, it is not continuous and hence not differentiable.

For (e) Not differentiable, Discontinuous, Partial derivatives defined

(because “not continuous” will mean it’s not differentiable, but I’m unsure of the Partial derivatives portion though because it appears the the partial derivatives are $0$ but I also have the feeling that the partial derivatives do not exist.)

For (f) Not differentiable, Continuous, Partial derivatives defined. This question appears to be similar to question (d)

I have already attempted these questions many times, but I keep answering these questions incorrectly. I know that I must be missing out on some parts especially when these are tricky questions which are not as simple it might seem. Could anyone help me please? Thanks!

- Equicontinuity and uniform convergence 2
- Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?
- Is a differentiable function always continuous?
- Find the value of : $\lim_{x \to \infty} \sqrt{4x^2 + 4} - (2x + 2)$
- Expressing the solutions of the equation $ \tan(x) = x $ in closed form.
- Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$
- Finding points on the parabola at which normal line passes through it
- When can we not treat differentials as fractions? And when is it perfectly OK?
- Show that $a - b \mid f(a) - f(b)$
- Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?

(a) it is a compostion of diferentiable functions, then it is differentiable, and contious and the partial derivative exist in $(0,0)$.

(b) It is continuous,and we have that the partial derivative are

$f_x(0,0)=\displaystyle\lim_{t\to 0}\frac{f(t,0)-f(0,0)}{t}=

\lim_{t\to 0}\frac{\sqrt{|t0|}-\sqrt{|0\times0|}}{t}=0$, and

$f_y(0,0)=\displaystyle\lim_{t\to 0}\frac{f(0,t)-f(0,0)}{t}=

\lim_{t\to 0}\frac{\sqrt{|t0|}-\sqrt{|0\times0|}}{t}=0$. however it is not differentiable since $\lim_{t\to0^+}\frac{f(t,t)-f(0,0)}{t}=1$ and $\lim_{t\to0^-}\frac{f(t,t)-f(0,0)}{t}=-1$,then it is not diferentiable.

(c) It is continuous because it is composition of continuous functions,but

$f_x(0,0)=\displaystyle\lim_{t\to 0^+}\frac{f(t,0)-f(0,0)}{t}=

\lim_{t\to 0^+}\frac{1 − \sin\sqrt{t^2 + 0^2}- (1 − \sin\sqrt{0})}{t}=-1$ but

$f_x(0,0)=\displaystyle\lim_{t\to 0^-}\frac{f(t,0)-f(0,0)}{t}=

\lim_{t\to 0^-}\frac{1 − \sin\sqrt{t^2 + 0^2}- (1 − \sin\sqrt{0})}{t}=1$

and the partial $f_x$ is not defined in $(0,0)$, analogously for $f_y(0,0)$, both does not exist.

Then it is not differentiable, because a differentiable function the elimites above should exist.

(d) it is not continuous, because $t>0$ then $(t,t)\to 0$ then $f(t,t)=1/2\neq 0=f(0,0)$. And it is not differentiable since it is not continuous. However

$f_x(0,0)=\displaystyle\lim_{t\to 0}\frac{f(t,0)-f(0,0)}{t}=

\lim_{t\to 0^+}\frac{\dfrac{t0}{t^2+0^2}-0}{t}=0$ and

$f_y(0,0)=\displaystyle\lim_{t\to 0}\frac{f(0,t)-f(0,0)}{t}=

\lim_{t\to 0^+}\frac{\dfrac{t0}{t^2+0^2}-0}{t}=0$.

(e) It is clearly not continuous, hence not differentiable at $(0,0)$, but

$f_x=\displaystyle\lim_{t\to0}\frac{f(x+t,y)-f(x,t)}{t}=0$ and

$f_y=\displaystyle\lim_{t\to0}\frac{f(x,y+t)-f(x,t)}{t}=0$, are defined in $(0,0)$

(f)It is not continuous since $\lim_{t\to 0}f(2t,t)=\lim_{t\to0}\dfrac{4t^2-t^2}{4t^2+t^2}=\frac{3}{5}\neq f(0,0)$, hence it is not differentiable in $(0,0)$.

$f_x(0,0)=\displaystyle\lim_{t\to0^+}\frac{f(x+t,y)-f(x,t)}{t}

=\lim_{t\to 0^+}\frac{\dfrac{t^2-0^2}{t^2+0^2}-0}{t}=+\infty

$ analogously for $f_y(0,0)$, both are note defined in $(0,0)$.

- Elementary set theory – prove or disprove question
- Is this a known algebraic identity?
- Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$
- Show that If $R$ is Euclidean domain then $R$ is a field
- Name for the embedding property
- How to show that the triangle is equilateral triangle?
- Locally Free Sheaves
- Familys of curves in z-plane depending on 1 parameter
- Yitang Zhang: Prime Gaps
- Weak topology is not metrizable: what's wrong with this proof?
- Bijection between $\mathbb{R}$ and $\mathbb{R}/\mathbb{Q}$
- Prove that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
- Decomposing a discrete signal into a sum of rectangle functions
- A bounded net with a unique limit point must be convergent
- The locus of two perpendicular tangents to a given ellipse