Problems
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Prove that $f(x)=1/x$ is continuous for $x\lt 0$.
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Prove that $f(x)=1/x$ is continous for $x\gt 0$.
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Prove that $x^2$ is continuous.
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Determine \[ \lim_{x\to 3}\,-10x^4+7x^2+300. \]
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If $f(x)$ is continuous and $M(x)$ is the maximum value of $f(x)$ in the closed interval $[x,a]$, then prove that \[ \lim_{x\to a^-}\,M(x)=f(a). \]
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If $f(x)$ is continuous and $m(x)$ is the minimum value of $f(x)$ in the closed interval $[x,a]$, then prove that \[ \lim_{x\to a^-}\,m(x)=f(a). \]
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If $f(x)$ is continuous and $M(x)$ is the maximum value of $f(x)$ in the closed interval $[a,x]$, then prove that \[ \lim_{x\to a^+}\,M(x)=f(a). \]
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If $f(x)$ is continuous and $m(x)$ is the minimum value of $f(x)$ in the closed interval $[a,x]$, then prove that \[ \lim_{x\to a^+}\,m(x)=f(a). \]
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Prove that if $f(x)$ is continuous and $K$ is a constant, then $f(x)+K$ is a continuous function.
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$ or $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(b)\lt 0\lt f(a)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.
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Prove that if $f(x)$ is continuous at $a$ then there is an open interval around $a$ for which $f(x)$ is bounded.
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Prove that if $f(x)$ is continuous on $[a,b]$ and $f(a)\lt 0\lt f(b)$, then there exists a number $K$ between $a$ and $b$ such that $f(K)=0$.
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Prove that if $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $f(g(a))$ then the function composition $(f\circ g)(x)$ is continuous at $a$.
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Prove that if $f(x)$ is continuous at $a$ then there is a closed interval around $a$ for which $f(x)$ is bounded.
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Prove that if $g(x)$ is continuous at $a$ and $f(x)$ is continuous at $f(g(a))$ then the function composition $(f\circ g)(x)$ is continuous at $a$.
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Prove that if $f(x)$ is continuous at $a$ and $f(a)\neq 0$ then $1/f(x)$ is continuous at $a$.
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Prove that if \[\lim_{h\to 0}\,f(a+h)=f(a)\] then $f(x)$ is continuous at $a$.
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Prove that if $f(x)$ is continuous at $a$ then \[\lim_{h\to 0}\,f(a+h)=f(a).\]
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Prove that $f(x)$ is continuous at $a$ if and only if \[\lim_{h\to 0}\,f(a+h)=f(a).\]
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Function Limits of Polynomials
