Question

If $f\left(x\right)=|lo{g}_{10}^x|\forall xϵ(0,\infty )$ then

• A. f(x) is continuous in $(0,\infty )$
• B. f(x) is differentiable in $(0,\infty )$
• C. f(x) is continuous but not differentiable in $(0,\infty )$
• D. None of these

Answer 1

Answer: (A), (C)

f(x) is continuous but not differentiable in $(0,\infty )$

Clearly, we see f(x) is continuous from $(0,\infty )$ but there is a sharp edge at 1 which means LHD and RHD

are finite but not equal. Therefore f(x) is continuous in $(0,\infty )$ but not differentiable throughout $(0,\infty )$. It

is only differentiable in $xϵ(0,1)\cup (1,\infty )$.