Word Count : 521

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1. If a function is differentiable at a point a, then it is continuous at a. True. By definition, differentiability gives $\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=f'(a)$ finite, which implies $\lim_{x\to a}(f(x)-f(a))=0$, hence $\lim_{x\to a}f(x)=f(a)$. 2. If a function is continuous at a, then it is differentiable at a. False. The standard counterexample is $f(x)=|x|$ at $a=0$. The function is continuous everywhere, but the derivative from the left is $-1$ and from the right is $1$, so it is not differentiable at $0$. 3. If $f'(x)=0$ for all $x\in(a,b)$, then $f$ is constant on $(a,b)$. True. By the Mean Value Theorem (MVT), for any $x _________ ____ ______ _____ ________ ________ __________ __________.
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