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.