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 February 16th, 2014, 08:30 PM #1 Newbie   Joined: Feb 2014 Posts: 6 Thanks: 1 continuity and partial derivatives of a function If $n\in \mathbb{N}$ and if $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ given by $f=0$ if $(x,y)=(0,0)$ and $f=\frac{(x-y)^{n}}{x^{2}+2y^{2}}$ if $(x,y)\neq (0,0)$ 1. Values ??for $n$ is $f$ continuously throughout $\mathbb{R}^{2}$? 2. Values ??for $n$ are the partial derivatives of $f$ throughout $\mathbb{R}^{2}$? 3. Values ??for $n$ is $f$ differentiable at all $\mathbb{R}^{2}$?
 February 21st, 2014, 01:11 PM #2 Senior Member   Joined: Dec 2012 Posts: 372 Thanks: 2 Re: continuity and partial derivatives of a function 1.) As the quotient of two differentiable functions, the given function is clearly continuous and differentiable everywhere but at $(0, 0)$ where the denominator is zero. We therefore invoke some analytical tools to judge continuity at $(0, 0)$. $f$ is not continuous for $n= 1$. Going along the direction of $x= y$, we get $f$ to approach zero in the limit as $(x, y) \rightarrow (0, 0)$. However, going along the direction of $x= -y$, we get $f$ to approach $\infty$. $f$ is not continuous for $n= 2$ either. Going along the direction of $x= y$, we get $f$ to approach zero in the limit as $(x, y) \rightarrow (0, 0)$. However going along the direction of $x= -y$, we get $f$ to approach $\frac{4}{3}$ in the limit as $(x, y) \rightarrow (0, 0)$. $f$ is continuous for $n \geq 3$. This is shown by using the binomial expansion of the expression in the numerator and observing that $\dfrac{|(x - y)^n|}{x^2 + 2y^2} \leq \dfrac{(|x |+ |y|)^n}{x^2 + y^2}$. Approaching $(0, 0)$ in the function at the right hand side of the inequality, we get zero invariably, so that the function on the left hand side must also approach zero. 2.) Where $f$ is differentiable, its partial derivatives exist and are continuous and vice versa. Wherever $f_x , f_y$ are continuous then $f$ must be continuous also. Hence on $\mathbb{R}^2 \backslash \{(0, 0)\}$, $f_x= \dfrac{(x-y)^{n-1}(nx^2 + 2ny^2 -2x^2 + 2xy)}{(x^2 + 2y^2)^2} \\ f_y = \dfrac{(x-y)^{n-1}(-nx^2 - 2ny^2 - 4xy + 4y^2)}{(x^2 + 2y^2)^2}$ and since $f$ is not continuous at $(0, 0)$for $n= 1$ and $n= 2$ then $f_x , f_y$ are both not continuous. These partial derivatives are again not continuous for $n= 3$, as we see approaching $(0, 0)$ in either case along the direction of $x= y$ and the direction of $x= -y$. The partial derivatives are continuous on $\mathbb{R}^2$ for $n \geq 4$ by using a similar argument of binomial expansion of terms in the numerator. Questions 2 and 3 have the same answer because continuity of the two partial derivatives $f_x , f_y$ is equivalent to differentiablity of the function $f$.

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