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Section 1
Introduction
Prove that the function $f(x)=5 x-3$ is continuous at $x=0$, at $x=-3$ and at $x=5$.
Examine the continuity of the function $f(x)=2 x^{2}-1$ at $x=3$.
Examine the following functions for continuity.(a) $f(x)=x-5$(b) $f(x)=\frac{1}{x-5}, x \neq 5$(c) $f(x)=\frac{x^{2}-25}{x+5}, x \neq-5$(d) $f(x)=|x-5|$
Prove that the function $f(x)=x^{n}$ is continuous at $x=n$, where $n$ is a positive integer.
Is the function $f$ defined by$$f(x)=\left\{\begin{array}{l}x, \text { if } x \leq 1 \\5, \text { if } x>1\end{array}\right.$$continuous at $x=0 ?$ At $x=1$ ? At $x=2$ ?
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{l}2 x+3, \text { if } x \leq 2 \\2 x-3, \text { if } x>2\end{array}\right.$$
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{cl}|x|+3, & \text { if } x \leq-3 \\-2 x, & \text { if }-3<x<3 \\6 x+2, & \text { if } x \geq 3\end{array}\right.$$
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{c}\frac{|x|}{x}, \text { if } x \neq 0 \\0, \quad \text { if } x=0\end{array}\right.$$
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}\frac{x}{|x|}, & \text { if } x<0 \\-1, & \text { if } x \geq 0\end{array}\right.$$
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}x+1, & \text { if } x \geq 1 \\x^{2}+1, & \text { if } x<1\end{array}\right.$$
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}x^{3}-3, & \text { if } x \leq 2 \\x^{2}+1, & \text { if } x>2\end{array}\right.$$
Find all points of discontinuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}x^{10}-1, & \text { if } x \leq 1 \\x^{2}, & \text { if } x>1\end{array}\right.$$
Is the function defined by$$f(x)=\left\{\begin{array}{ll}x+5, & \text { if } x \leq 1 \\x-5, & \text { if } x>1\end{array}\right.$$a continuous function?
Discuss the continuity of the function $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}3, & \text { if } 0 \leq x \leq 1 \\4, & \text { if } 1<x<3 \\5, & \text { if } 3 \leq x \leq 10\end{array}\right.$$
Discuss the continuity of the function $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}2 x, & \text { if } x<0 \\0, & \text { if } 0 \leq x \leq 1 \\4 x, & \text { if } x>1\end{array}\right.$$
Discuss the continuity of the function $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}-2, & \text { if } x \leq-1 \\2 x, & \text { if }-1<x \leq 1 \\2, & \text { if } x>1\end{array}\right.$$
Find the relationship between $a$ and $b$ so that the function $f$ defined by$$f(x)=\left\{\begin{array}{ll}a x+1, & \text { if } x \leq 3 \\b x+3, & \text { if } x>3\end{array}\right.$$is continuous at $x=3$.
For what value of $\lambda$ is the function defined by$$f(x)=\left\{\begin{array}{ll}\lambda\left(x^{2}-2 x\right), & \text { if } x \leq 0 \\4 x+1, & \text { if } x>0\end{array}\right.$$continuous at $x=0$ ? What about continuity at $x=1$ ?
Show that the function defined by $g(x)=x-[x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less than or equal to $x$.
Is the function defined by $f(x)=x^{2}-\sin x+5$ continuous at $x=\pi$ ?
Discuss the continuity of the following functions:(a) $f(x)=\sin x+\cos x$(b) $f(x)=\sin x-\cos x$(c) $f(x)=\sin x \cdot \cos x$
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Find all points of discontinuity of $f$, where$$f(x)=\left\{\begin{array}{ll}\frac{\sin x}{x}, & \text { if } x<0 \\x+1, & \text { if } x \geq 0\end{array}\right.$$
Determine if $f$ defined by$$f(x)=\left\{\begin{array}{ll}x^{2} \sin \frac{1}{x}, & \text { if } x \neq 0 \\0, & \text { if } x=0\end{array}\right.$$is a continuous function?
Examine the continuity of $f$, where $f$ is defined by$$f(x)=\left\{\begin{array}{ll}\sin x-\cos x, & \text { if } x \neq 0 \\-1, & \text { if } x=0\end{array}\right.$$
Find the values of $k$ so that the function $f$ is continuous at the indicated point in exercises.$$f(x)=\left\{\begin{array}{ll}\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\3, & \text { if } x=\frac{\pi}{2}\end{array} \text { at } x=\frac{\pi}{2}\right.$$
Find the values of $k$ so that the function $f$ is continuous at the indicated point in exercises.$$f(x)=\left\{\begin{array}{ll}k x^{2}, & \text { if } x \leq 2 \\3, & \text { if } x>2\end{array} \quad \text { at } x=2\right.$$
Find the values of $k$ so that the function $f$ is continuous at the indicated point in exercises.$$f(x)=\left\{\begin{array}{ll}k x+1, & \text { if } x \leq \pi \\\cos x, & \text { if } x>\pi\end{array} \quad \text { at } x=\pi\right.$$
Find the values of $k$ so that the function $f$ is continuous at the indicated point in exercises.$$f(x)=\left\{\begin{array}{ll}k x+1, & \text { if } x \leq 5 \\3 x-5, & \text { if } x>5\end{array} \quad \text { at } x=5\right.$$
Find the values of $a$ and $b$ such that the function defined by$$f(x)=\left\{\begin{array}{ll}5, & \text { if } x \leq 2 \\a x+b, & \text { if } 2<x<10 \\21, & \text { if } x \geq 10\end{array}\right.$$is a continuous function.
Show that the function defined by $f(x)=\cos \left(x^{2}\right)$ is a continuous function.
Show that the function defined by $f(x)=|\cos x|$ is a continuous function.
Examine that $\sin |x|$ is a continuous function.
Find all the points of discontinuity of $f$ defined by $f(x)=|x|-|x+1|$.