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

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- Oct 22nd 2008, 02:38 PMiz1hpcontinuous, not differentiable
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 - Oct 22nd 2008, 02:51 PMskeeter
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. - Oct 22nd 2008, 02:56 PMiz1hp
thanks skeeter!