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Let $U$ be an open set on ${\mathbb R}^{n}$ (but $U$ is not an empty set), $\textbf{p}\in{U}$, and $f:U\to \mathbb R$ is continuously-differentiable on $U$. Then, it is known that, "the function $f$ can be differentiable for all $\textbf{q}\in U$." (See Spivac)

And I know that, there is a function $f$ such that it is differentiable at $\textbf{p}$ but, for any $r> 0$, $f$ is not differentiable (and continuously differentiable) on $U_{\textbf{p}} (r)$ . Here, $U_{\textbf{p}} (r)$ is an open ball of radius $r$ centered on $\textbf{p}$.
For example, if $f:{\mathbb R}^{2}\to \mathbb R$ is defined as follows, $f$ is differentiable at $\textbf{0}$,, but is not differentiable (and not continuous) at any other point. Here $\mathbb Q$ is the set of all rational numbers, and $U_{\textbf{p}} (r)$ is an open ball of radius $r$ centered on $\textbf{p}$.

$f(x,y):=\left\{ \begin{array}{rr} 0, & (x,y)\in \mathbb Q^{2} \\ x^2 + y^2, & (x,y)\notin \mathbb Q^{2} \\ \end{array} \right.$

Therefore, there is at least one function that does not have a continuously-differentiable region, even if it can be differentiable at one point. But I cannot imagine whether are there any functions that are differentiable on $U$ but not continuously-differentiable.

My question
Let $U$ be an open set of $\mathbb R^n$ (but is not an empty set), and $\ \textbf{p}\in U $.
Then, are there any functions $f:U\to \mathbb R$ such that, $f$ is differentiable on $U$, but for any $r> 0$, $f$ is not continuously-differentiable on $U_{\textbf{p}} (r)$ ?
If so, give an example. If not, please explain why.
Here, $U_{\textbf{p}} (r)$ is an open ball of radius $r$ centered on $\textbf{p}$.

Here, the definitions of differentiable and continuously differentiable are as follows.

Def1 (Differentiable at $\textbf{p}$)
Let $U$ be an open set (but not empty set) of ${\mathbb R}^{n}$, $\textbf {p} \in \mathbb R^n$, and $f$ is a function whose domain is $U$. At this time, $f$ is differentiable at $\textbf{p}$ iff the following is satisfied.
${\exists} A:{\mathbb R}^{n}\to \mathbb R$: a linear map such that
  $${\lim}_{\textbf{x}\to\textbf{p}}\frac{|f(\textbf{x}) - A(\textbf{x}-\textbf{p}) - f(\textbf{p})|}{|\textbf{x}-\textbf{p}|} = 0$$


Def2 (Differentiable on $\textbf{U}$)
Let $U$ be an open set (but not empty sets) of ${\mathbb R}^{n}$, and $f$ is a function which domain is $U$.
At this time, $f$ is differentiable at $U$ iff "for all $\textbf{q}\in{\mathbb R}^{n}$, $f$ is differentiable at $\textbf{q}$".


Def3 (Continuously-differentiable on $U$)
Let $U$ be an open set (but is not empty sets) of ${\mathbb R}^{n}$, and $f$ is a function which domain is $U$. At this time, $f:U\to \mathbb R$ is continuously-differentiable on $U$ iff

  • $f$ is partially differentiable for all direction, ${x}_{1}, {x}_{2}, ..., {x}_{n}$ (that mean, we can define $\frac{\partial f}{\partial{x}_{1}}, \cdots\frac{\partial f} {\partial{x}_{n}} $ on $U$). and,
  • $\frac{\partial f} {\partial{x}_{1}}, \cdots\frac{\partial f} {\partial{x}_{n}} $ are continuous on $U$.

P.S.
I'm not very good at English, so I'm sorry if I have some impolite or unclear expressions.


Post-hoc Note:【Verification of the function taught by Thomas Shelby】
The following are the confirmation that the following function $f$ meets my requirement (Is it correct as proof?):

$f(x,y)=\begin{cases}(x^2+y^2)\sin\left(\frac1 {\sqrt{x^2+y^2}}\right),&(x,y)\neq 0\\0,&(x,y)=0\end{cases}.$

My proof:
${\lim}_{\|\textbf{x}\|\to 0} \frac{f(\textbf{x}) - f(\textbf{0})}{\|\textbf{x}\|}= {\lim}_{\|\textbf{x}\|\to 0} \frac{{\|\textbf{x}\|}^{2}\sin(1/ \|\textbf{x}\| - 0)}{\|\textbf{x}\|}= $ ${\lim}_{\|\textbf{x}\|\to 0} \|\textbf{x}\|\sin(1/ \|\textbf{x}\|) = 0$
Therefore, the $f$ is differntiable at $(0,0)$ and $Jf(0,0)=(0,0)$.

On the other hand, for $\textbf{x}\neq\textbf{0}$,
Let $g$ and $h$ be $g(x,y):=\sqrt{{x}^2 + {y}^2}\ $ and $\ h(t):={t}^{2}\sin(1/t)$ (for $t\neq 0$) respectively, then

$$\frac{d\sqrt{t}}{dt} = \frac{1}{2\sqrt{t}} $$ and, $$(J\|\textbf{x}\|^2)(x,y) = (2x,2y) ,$$ Therefore, $$(Jg)(x,y) = \left(\frac{x}{\|\textbf{x}\|} , \frac{y}{\|\textbf{x}\|}\right)\quad (\textrm{for all $\textbf{x}\neq\textbf{0}\ $}),$$ and $$\ \frac{d\sin(1/t)}{dt} = -\frac{\cos(1/t)}{t^2}\ \ \ (\textrm{at $t\neq 0$}).$$ Therefore, $$\frac{dh}{dt} ={t}^{2}\frac{d\sin(1/t)}{dt} + 2t\sin(1/t) = -\cos(1/t) + 2t\sin(1/t).$$

Therefore, at $\textbf{x}\neq \textbf{0}$,
$$Jf(x,y) = \left(\left.\frac{dh}{dt}\right|_{t=||\textbf{x}||}\right)(Jg)(x,y) = (-\cos(1/||\textbf{x}||) + 2t\sin(1/||\textbf{x}||))\left(\frac{x}{||\textbf{x}||} , \frac{y}{||\textbf{x}||}\right).$$

Therefore,
$$\frac{\partial f}{\partial x} = -\frac{x\cos(1/||\textbf{x}||)}{||\textbf{x}||} + 2x\sin(1/||\textbf{x}||),\,\textbf{x}\neq\textbf{0}$$and $$\frac{\partial f}{\partial y} = -\frac{y\cos(1/||\textbf{x}||)}{||\textbf{x}||} + 2y\sin(1/||\textbf{x}||),\,\textbf{x}\neq\textbf{0}.$$

However, both $\dfrac{x\cos(1/||\textbf{x}||)}{||\textbf{x}||} $ and $\dfrac{y\cos(1/||\textbf{x}||)}{||\textbf{x}||} $ do not get converted at $(0,0)$.

So, both $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are not continuous at $(0,0)$. ■

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  • $\begingroup$ I am guessing you are looking for something less continuously differentiable that the one here math.stackexchange.com/q/1391544/27978? $\endgroup$
    – copper.hat
    Aug 30 '19 at 4:07
  • $\begingroup$ Related (maybe, I'm not sure I exactly understand what your question is): math.stackexchange.com/questions/112067/… $\endgroup$ Aug 30 '19 at 6:35
  • $\begingroup$ @copper.hat: Thank you for your comment. The answer from Thomas Shelby satisfies my requirement. It was unexpected that there was such a simple example. Actually, I didn't realize that, we could make "multivariable functions that is continuously-diffentiable" from "univariate function that is differentiable but not continuously-diffentiable." But, as you say, his answer looks equivalent to your link. $\endgroup$ Aug 30 '19 at 10:21
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    $\begingroup$ @BlueVarious Overall, the proof is good. But I think $(Jg)(x,y) = \left(\frac{x}{\|\textbf{x}\|} , \frac{y}{\|\textbf{x}\|}\right)$. So you should swap $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial x}$. $\endgroup$
    – Cop 663
    Aug 30 '19 at 13:41
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An example for $n = 1$ from the theory of random walks. Let $f$ be a (-n everywhere) discontinuous Lebesgue measurable function on $\mathbb{R}$. Here's an example with $f$ bounded by $1$, just showing the part $x \in [-3,3]$. (Note that I have only barely subsampled the graph in this interval. If I were to fully sample it, this finite resolution representation would almost surely appear to be a solid rectangle of points of the graph. Actually produced by generating $10^6$ uniformly distributed reals in $[-1,1]$ assigned to evenly spaced abscissae, then plotting a subsample of size $10^4$.)

An almost surely nowhere continuous function

This function is almost surely nowhere continuous (as any open interval almost surely contains points of heights arbitrarily close to $-1$ and $1$). The integral of this function, $$ \int_{0}^x \; f(t) \,\mathrm{d}t $$ is differentiable, but there's no hope of continuous differentiability. Graph of the integral (actually, Riemann sum approximations using $10^6$ intervals in $[-3,3]$):

integral

Picking a different instance of a bounded by $1$ discontinuous Lebesgue measurable function on $\mathbb{R}$ and integrating it the same way, we can graph the integral.

integral 2

These are almost everywhere differentiable by construction (by Lebesgue's differentiation theorem); we know the derivative is $f$. (The theorem generalizes to $n > 1$ and the integral to $\int_{[0,x_1]\times [0,x_2] \times \cdots \times [0,x_n]} \; f(t) \,\mathrm{d}t$ where we understand the intervals to be $[0,a]$ when $0 \leq a$ and $[a,0]$ when $a < 0$.) In some way, "most" functions are everywhere discontinuous messes, so "most" functions can be integrated to a differentiable, but not continuously differentiable, function.

(This construction can be iterated to get a function that is several times continuously differentiable, but whose "last" derivative is not continuous.)

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  • $\begingroup$ Thank you for your answer. Is it correct to understand that "your function $f$ is differentiable at any point, but cannot be continuously-differentiable almost everywhere"? $\endgroup$ Aug 30 '19 at 10:38
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    $\begingroup$ @BlueVarious : My $f$ is the derivative function. We have chosen $f$ to be everywhere discontinuous. Then $\int f$ is a continuous function with a known derivative and that known derivative is everywhere discontinuous by construction. (It is also almost everywhere discontinuous since the empty set has measure zero, but this is a weaker statement.) $\endgroup$ Aug 30 '19 at 14:31
  • $\begingroup$ Thank you for your comment. So, your function F=∫f is differentiable at all points, but not continuously-differntiable at all points? $\endgroup$ Aug 31 '19 at 11:34
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    $\begingroup$ @BlueVarious : Yes. $\endgroup$ Aug 31 '19 at 15:45
  • $\begingroup$ Thank you for your answer. Thank you for teaching me a very “morbid” but very interesting function. $\endgroup$ Aug 31 '19 at 16:31
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Consider $f:\mathbb R^2\to \mathbb R $ defined by $$f(x,y)=\begin{cases}(x^2+y^2)\sin\left(\frac1 {\sqrt{x^2+y^2}}\right),&(x,y)\neq 0\\0,&(x,y)=0\end{cases}.$$ Then $f $ is differentiable everywhere but $\dfrac{\partial f}{\partial x}(x,y)$ and $\dfrac{\partial f}{\partial y}(x,y)$ are not continuous at $(0,0)$.


A detailed calculation for the above example can be found in the book Functions of Several Real Variables.

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