Problem Set - Differentiability

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Differentiability

	Let $a,b\in ℝ$ and let $f:[a,b]\to ℝ$ be a function. Let $c\in [a,b]$ and carefully define $f\prime \left(c\right)$. Prove that if $f:[a,b]\to ℝ$ and $g:[a,b]\to ℝ$ are functions then $\left(fg\right)\prime \left(c\right)=f\left(c\right)g\prime \left(c\right)+f\prime \left(c\right)g\left(c\right)$, whenever $f\prime \left(c\right)$ and $g\prime \left(c\right)$ exist.
	Let $f:{ℝ}_{>0}\to ℝ$ be such that $f$ is differentiable at $x=1$ and if $x,y\in {ℝ}_{>0}$ then $f\left(xy\right)=f\left(x\right)+f\left(y\right)$. Show that (a)   if $c\in {ℝ}_{>0}$ then $f$ is differentiable at $x=c$, (b)   if $c\in {ℝ}_{>0}$ then $f\prime \left(c\right)=f\prime \left(1\right)/c$, (c)   Show that $f$ is infinitely differentiable.
	Let $f:ℝ\to ℝ$ be such that $f$ is differentiable at $x=0$ and if $x,y\in ℝ$ then $f(x+y)=f\left(x\right)f\left(y\right)$. Show that (a)   if $c\in ℝ$ then $f$ is differentiable at $x=c$, (b)   if $c\in {ℝ}_{>0}$ then $f\prime \left(c\right)=f\prime \left(0\right)f\left(c\right)$, (c)   Show that $f$ is infinitely differentiable.
	Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}-x^2,& \text{if}x\le 0,\hfill \\ x,& \text{if}x>0.\hfill \end{array}$ Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$?
	Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}-x^2,& \text{if}x\le 0,\hfill \\ x^3,& \text{if}x>0.\hfill \end{array}$ Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$?
	Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x}{x},& \text{if}x<0,\hfill \\ 1+x^2,& \text{if}x\ge 0.\hfill \end{array}$ Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$?
	Let $a,b\in ℝ$ and assume that $f:[a,b)\to ℝ$ is differentiable on $(a,b)$ and continuous on $[a,b)$. Assume that the limit $\underset{x\to a+}{\lim }f\prime \left(x\right)=L$ exists. Prove that the right derivative $f_{+}\prime \left(a\right)$ exists and that $f_{+}\prime \left(a\right)=L$.
	Let $a,b\in ℝ$ and assume that $f:(a,b)\to ℝ$ is differentiable at $c$. Show that $\underset{h\to 0+}{\lim }{\displaystyle \frac{f(c+h)-f(c-h)}{2h}}$ exists and equals $f\prime \left(c\right)$. Is the converse true?
	Prove that $\frac{d}{dx}\sqrt{x}={\displaystyle \frac{1}{2\sqrt{x}}}$.
	Prove that $\frac{d}{dx}\arcsin x={\displaystyle \frac{1}{\sqrt{1-x^2}}}$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)