Let $f$ be the the function given by $f(x)=|x|$. Which of the following statements about $f$ are true?

1. $\hskip{1em }f$ is continuous at $x=0$.
2. $\hskip{1em }f$ is differentiable at $x=0$
3. $\hskip{1em}f$ has an absolute maximum at $x=0$.

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The absolute value function is continous at all points. We can draw the the entire graph without lifting our pencil. $f(0)$ is also defined.

The function is not differentiable. Approaching from the left, we have a slope of $-1$. From the right, the slope is $1$.

$$\lim\limits_{x\to 0^-} f'(x) \neq \lim\limits_{x\to 0^+} f'(x)$$ $$-1 \neq 1$$

Furthermore, the graph has an absolute minimum at $x=0$, not maximum.