1. ## Continuity

Let f(x) be a continuous function in R. If the limit $\lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2}$ exists and it is a real number, find the value of $\alpha$ so that the function g(x) = $\lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2} \quad for \quad x \ne 2$
$\alpha \quad for \quad x = 2$

is continuous

I am trying to prove that g(x) is differentiable, because when a function is differentiable is continuous as well, but I can't. Can anyone help?

2. ## Re: Continuity

Originally Posted by ManosG
Let f(x) be a continuous function in R. If the limit $\lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2}$ exists and it is a real number, find the value of $\alpha$ so that the function g(x) = $\lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2} \quad for \quad x \ne 0$
$\alpha \quad for \quad x = 2$ is continuous

I think that you mean $g(x) = \left\{ {\begin{array}{rl} {\dfrac{{|f(x - 2)| + |x - 2|}}{{x - 2}},}&{x \ne 2} \\ {\alpha ,}&{x = 2} \end{array}} \right.$

If that is correct then $\lim_{x\to\2} \frac{|f(x-2)| + |x-2|}{x-2}=L=\alpha.$

3. ## Re: Continuity

Only that? We can't find a specific value?? Thanks for your help.