GAIN AN ADVANTAGE
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Lesson Specific Problems
- If $f(x)$ is continuous and $m(x)$ is the minimum value of $f(x)$ in the closed interval $[x,a]$, then prove that \[ \lim_{x\to a^-}\,m(x)=f(a). \]
- If $f(x)$ is continuous and $M(x)$ is the maximum value of $f(x)$ in the closed interval $[a,x]$, then prove that \[ \lim_{x\to a^+}\,M(x)=f(a). \]
- If $f(x)$ is continuous and $m(x)$ is the minimum value of $f(x)$ in the closed interval $[a,x]$, then prove that \[ \lim_{x\to a^+}\,m(x)=f(a). \]
- Prove that if $f(x)$ is continuous and $K$ is a constant, then $f(x)+K$ is a continuous function.
- 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 $f(K)=0$.
