Problem Set - Where is a Function Continuous?

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

Where is a Function Continuous?

	For which values of $x$ is the function $f\left(x\right)=x^2+3x+4$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-x-6}{x-3}\text{,}& \text{if }x\ne 3\text{,}\\ 5\text{,}& \text{if }x=3\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 3x}{x}\text{,}& \text{if }x\ne 0\text{,}\\ 1\text{,}& \text{if }x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{1-\cos x}{x^2}\text{,}& \text{if }x\ne 0\text{,}\\ 1\text{,}& \text{if }x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	Determine the value of $k$ for which the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 2x}{5x}\text{,}& \text{if }x\ne 0\text{,}\\ k\text{,}& \text{if }x=0\text{,}\end{array}\right.$$ contunuous at $x=0$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}x-1\text{,}& \text{if }1\le x<2\text{,}\\ 2x-3\text{,}& \text{if }2\le x\le 3\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\cos x\text{,}& \text{if }x\ge 0\text{,}\\ -\cos x\text{,}& \text{if }x<0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\sin \left(1/x\right)\text{,}& \text{if }x\ne 0\text{,}\\ 0\text{,}& \text{if }x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	Determine the value of $a$ for which the function $$f\left(x\right)=\left\{\begin{array}{ll}ax+5\text{,}& \text{if }x\le 2\text{,}\\ x-1\text{,}& \text{if }x>2\text{,}\end{array}\right.$$ contunuous at $x=2$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}1+x^2\text{,}& \text{if }0\le x\le 1\text{,}\\ 2-x\text{,}& \text{if }x>1\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $f\left(x\right)=2x-\left|x\right|$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	Find the value of $a$ for which the function $$f\left(x\right)=\left\{\begin{array}{ll}2x-1\text{,}& \text{if }x<2\text{,}\\ a\text{,}& \text{if }x=2\text{,}\\ x+1\text{,}& \text{if }x>2\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\left|x-a\right|}{x-a}\text{,}& \text{if }x\ne a\text{,}\\ 1\text{,}& \text{if }x=a\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x-\left|x\right|}{2}\text{,}& \text{if }x\ne 0\text{,}\\ 2\text{,}& \text{if }x=0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\sin x\text{,}& \text{if }x<0\text{,}\\ \mathrm{x}\text{,}& \text{if }x\ge 0\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For which values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^n-1}{x-1}\text{,}& \text{if }x\ne 1\text{,}\\ \mathrm{n}\text{,}& \text{if }x=1\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	Explain how you know $f\left(x\right)=\cos x$ is continuous for all values of $x$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	Explain how you know $f\left(x\right)=\cos \left|x\right|$ is continuous for all values of $x$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	Explain how you know $f\left(x\right)=⌊x⌋$ is continuous for all values of $x$. Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For what values of $x$ is the function $$f\left(x\right)=\left\{\begin{array}{ll}{x}^3-x^2+2x-2\text{,}& \text{if }x\ne 1\text{,}\\ 4\text{,}& \text{if }x=1\text{,}\end{array}\right.$$ contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.
	For what values of $x$ is the function $f\left(x\right)=\left|x\right|+\left|x-1\right|\text{, }-1\le x\le 2$, contunuous? Justiffy your answer with limits if necessary and draw a graph of the function to illustrate your answer.

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)