# No minimum at the origin but a minimum along all lines

We look here at an example, from the Italian mathematician Giuseppe Peano of a real function $$f$$ defined on $$\mathbb{R}^2$$. $$f$$ is having a local minimum at the origin along all lines passing through the origin, however $$f$$ does not have a local minimum at the origin as a function of two variables.

The function $$f$$ is defined as follows
$\begin{array}{l|rcl} f : & \mathbb{R}^2 & \longrightarrow & \mathbb{R} \\ & (x,y) & \longmapsto & f(x,y)=3x^4-4x^2y+y^2 \end{array}$ One can notice that $$f(x, y) = (y-3x^2)(y-x^2)$$. In particular, $$f$$ is strictly negative on the open set $$U=\{(x,y) \in \mathbb{R}^2 \ : \ x^2 < y < 3x^2\}$$, vanishes on the parabolas $$y=x^2$$ and $$y=3 x^2$$ and is strictly positive elsewhere. Consider a line $$D$$ passing through the origin. If $$D$$ is different from the coordinate axes, the equation of $$D$$ is $$y = \lambda x$$ with $$\lambda > 0$$. We have $f(x, \lambda x)= x^2(\lambda-3x)(\lambda -x).$ For $$x \in (-\infty,\frac{\lambda}{3}) \setminus \{0\}$$, $$f(x, \lambda x) > 0$$ while $$f(0,0)=0$$ which proves that $$f$$ has a local minimum at the origin along the line $$D \equiv y – \lambda x=0$$. Along the $$x$$-axis, we have $$f(x,0)=3 x^ 4$$ which has a minimum at the origin. And finally, $$f$$ also has a minimum at the origin along the $$y$$-axis as $$f(0,y)=y^2$$.

However, along the parabola $$\mathcal{P} \equiv y = 2 x^2$$ we have $$f(x,2 x^2)=-x^4$$ which is strictly negative for $$x \neq 0$$. As $$\mathcal{P}$$ is passing through the origin, $$f$$ assumes both positive and negative values in all neighborhood of the origin.

This proves that $$f$$ does not have a minimum at $$(0,0)$$.

# Counterexamples around Fubini’s theorem

We present here some counterexamples around the Fubini theorem.

We recall Fubini’s theorem for integrable functions:
let $$X$$ and $$Y$$ be $$\sigma$$-finite measure spaces and suppose that $$X \times Y$$ is given the product measure. Let $$f$$ be a measurable function for the product measure. Then if $$f$$ is $$X \times Y$$ integrable, which means that $$\displaystyle \int_{X \times Y} \vert f(x,y) \vert d(x,y) < \infty$$, we have $\int_X \left( \int_Y f(x,y) dy \right) dx = \int_Y \left( \int_X f(x,y) dx \right) dy = \int_{X \times Y} f(x,y) d(x,y)$ Let's see what happens when some hypothesis of Fubini's theorem are not fulfilled. Continue reading Counterexamples around Fubini’s theorem

# Differentiability of multivariable real functions (part2)

Following the article on differentiability of multivariable real functions (part 1), we look here at second derivatives. We consider a function $$f : \mathbb R^n \to \mathbb R$$ with $$n \ge 2$$.

Schwarz’s theorem states that if $$f : \mathbb R^n \to \mathbb R$$ has continuous second partial derivatives at any given point in $$\mathbb R^n$$, then for $$(a_1, \dots, a_n) \in \mathbb R^n$$ and $$i,j \in \{1, \dots, n\}$$:
$\frac{\partial^2 f}{\partial x_i \partial x_j}(a_1, \dots, a_n)=\frac{\partial^2 f}{\partial x_j \partial x_i}(a_1, \dots, a_n)$

### A function for which $$\frac{\partial^2 f}{\partial x \partial y}(0,0) \neq \frac{\partial^2 f}{\partial y \partial x}(0,0)$$

We consider:
$\begin{array}{l|rcl} f : & \mathbb R^2 & \longrightarrow & \mathbb R \\ & (0,0) & \longmapsto & 0\\ & (x,y) & \longmapsto & \frac{xy(x^2-y^2)}{x^2+y^2} \text{ for } (x,y) \neq (0,0) \end{array}$ Continue reading Differentiability of multivariable real functions (part2)

# Differentiability of multivariable real functions (part1)

This article provides counterexamples about differentiability of functions of several real variables. We focus on real functions of two real variables (defined on $$\mathbb R^2$$). $$\mathbb R^2$$ and $$\mathbb R$$ are equipped with their respective Euclidean norms denoted by $$\Vert \cdot \Vert$$ and $$\vert \cdot \vert$$, i.e. the absolute value for $$\mathbb R$$.

We recall some definitions and theorems about differentiability of functions of several real variables.

Definition 1 We say that a function $$f : \mathbb R^2 \to \mathbb R$$ is differentiable at $$\mathbf{a} \in \mathbb R^2$$ if it exists a (continuous) linear map $$\nabla f(\mathbf{a}) : \mathbb R^2 \to \mathbb R$$ with $\lim\limits_{\mathbf{h} \to 0} \frac{f(\mathbf{a}+\mathbf{h})-f(\mathbf{a})-\nabla f(\mathbf{a}).\mathbf{h}}{\Vert \mathbf{h} \Vert} = 0$

Definition 2 Let $$f : \mathbb R^n \to \mathbb R$$ be a real-valued function. Then the $$\mathbf{i^{th}}$$ partial derivative at point $$\mathbf{a}$$ is the real number
\begin{align*}
\frac{\partial f}{\partial x_i}(\mathbf{a}) &= \lim\limits_{h \to 0} \frac{f(\mathbf{a}+h \mathbf{e_i})- f(\mathbf{a})}{h}\\
&= \lim\limits_{h \to 0} \frac{f(a_1,\dots,a_{i-1},a_i+h,a_{i+1},\dots,a_n) – f(a_1,\dots,a_{i-1},a_i,a_{i+1},\dots,a_n)}{h}
\end{align*} For two real variable functions, $$\frac{\partial f}{\partial x}(x,y)$$ and $$\frac{\partial f}{\partial y}(x,y)$$ will denote the partial derivatives.

Definition 3 Let $$f : \mathbb R^n \to \mathbb R$$ be a real-valued function. The directional derivative of $$f$$ along vector $$\mathbf{v}$$ at point $$\mathbf{a}$$ is the real $\nabla_{\mathbf{v}}f(\mathbf{a}) = \lim\limits_{h \to 0} \frac{f(\mathbf{a}+h \mathbf{v})- f(\mathbf{a})}{h}$ Continue reading Differentiability of multivariable real functions (part1)

# Continuity of multivariable real functions

This article provides counterexamples about continuity of functions of several real variables. In addition the article discusses the cases of functions of two real variables (defined on $$\mathbb R^2$$ having real values. $$\mathbb R^2$$ and $$\mathbb R$$ are equipped with their respective Euclidean norms denoted by $$\Vert \cdot \Vert$$ and $$\vert \cdot \vert$$, i.e. the absolute value for $$\mathbb R$$.

We recall that a function $$f$$ defined from $$\mathbb R^2$$ to $$\mathbb R$$ is continuous at $$(x_0,y_0) \in \mathbb R^2$$ if for any $$\epsilon > 0$$, there exists $$\delta > 0$$, such that $$\Vert (x,y) -(x_0,y_0) \Vert < \delta \Rightarrow \vert f(x,y) - f(x_0,y_0) \vert < \epsilon$$. Continue reading Continuity of multivariable real functions