Question

Check the function f(x)=[2x-1, 0≤x≤1
[1, x>1
for continuity and differentiability at x =1

Answer 1

$\lim_{x\to 1^-}f(x)=\lim_{x\to 1^+}f(x)=1\:\;so,\;f(x)\;is\;continuous\;at\;x=1\\ \lim_{\to 1^-}\frac{f(x)-f(1)}{x-1}=\lim_{\to 1^-}\frac{2(x-1)}{x-1}=2, \lim_{\to 1^+}\frac{f(x)-f(1)}{x-1}=0\\ so,\;f(x)\;is\;not\;differentiable\;at\;x=1.$