Last updates: 7 December 2009

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 $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}3x}{x}\text{,}& \text{if}x\ne 0\text{,}\\ 1\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}\frac{1-\mathrm{cos}x}{{x}^{2}}\text{,}& \text{if}x\ne 0\text{,}\\ 1\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}2x}{5x}\text{,}& \text{if}x\ne 0\text{,}\\ k\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}x-1\text{,}& \text{if}1\le x2\text{,}\\ 2x-3\text{,}& \text{if}2\le x\le 3\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}\mathrm{cos}x\text{,}& \text{if}x\ge 0\text{,}\\ -\mathrm{cos}x\text{,}& \text{if}x0\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}\mathrm{sin}\left(1/x\right)\text{,}& \text{if}x\ne 0\text{,}\\ 0\text{,}& \text{if}x=0\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}ax+5\text{,}& \text{if}x\le 2\text{,}\\ x-1\text{,}& \text{if}x2\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}1+{x}^{2}\text{,}& \text{if}0\le x\le 1\text{,}\\ 2-x\text{,}& \text{if}x1\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}2x-1\text{,}& \text{if}x2\text{,}\\ a\text{,}& \text{if}x=2\text{,}\\ x+1\text{,}& \text{if}x2\text{,}\end{array}\right.$$ 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 $$f\left(x\right)=\left\{\begin{array}{ll}\mathrm{sin}x\text{,}& \text{if}x0\text{,}\\ \mathrm{x}\text{,}& \text{if}x\ge 0\text{,}\end{array}\right.$$ 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 $f\left(x\right)=\mathrm{cos}x$ 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 $f\left(x\right)=\mathrm{cos}\left|x\right|$ 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 $f\left(x\right)=\u230ax\u230b$ 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. |

[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)