# continuous, not differentiable

• October 22nd 2008, 02:38 PM
iz1hp
continuous, not differentiable
Given the function:

$

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
• October 22nd 2008, 02:51 PM
skeeter
continuous at x = k ...

$f(k) = b$ ... $f(k)$ exists

$\lim_{x \to k^-} f(x) = \lim_{x \to k^+} f(x) = b = f(k)$

therefore, f(x) is continuous at x = k

differentiability at x = k ...

$\lim_{x \to k} \frac{f(x) - f(k)}{x - k} =$

$\lim_{x \to k} \frac{a|x-k|+b - b}{x - k} =$

$\lim_{x \to k} \frac{a|x-k|}{x - k} =$

$a$ if $x \to k^+$

$-a$ if $x \to k^-$

since the limit does not exist, f(x) is not differentiable at x = k.
• October 22nd 2008, 02:56 PM
iz1hp
thanks skeeter!