Given the function:
$\displaystyle
f(x) = a |x - k| + b
$
where a, b and k are constants, show that f(x) is continuous but not differentiable at x = k
Thank you to everyone who helps <3
continuous at x = k ...
$\displaystyle f(k) = b$ ... $\displaystyle f(k)$ exists
$\displaystyle \lim_{x \to k^-} f(x) = \lim_{x \to k^+} f(x) = b = f(k) $
therefore, f(x) is continuous at x = k
differentiability at x = k ...
$\displaystyle \lim_{x \to k} \frac{f(x) - f(k)}{x - k} =$
$\displaystyle \lim_{x \to k} \frac{a|x-k|+b - b}{x - k} =$
$\displaystyle \lim_{x \to k} \frac{a|x-k|}{x - k} =$
$\displaystyle a$ if $\displaystyle x \to k^+$
$\displaystyle -a$ if $\displaystyle x \to k^-$
since the limit does not exist, f(x) is not differentiable at x = k.