Question

Show that the function f(x) = 2x - |x| is continuous at x = 0

Answer 1

${\displaystyle \text{L}{\left[\text{f}\left(x\right)\right]}_{x=0}=\underset{x\to 0^{-}}{lim}\text{f}\left(x\right)=\underset{h\to 0}{lim}\text{f}(0-\text{h})}$

${\displaystyle =\underset{h\to 0}{lim}\text{f}(-\text{h})}$

${\displaystyle =\underset{h\to 0}{lim}2(-\text{h})-\left|\text{h}\right|}$

${\displaystyle =\underset{h\to 0}{lim}-2\text{h}-\text{h}}$

${\displaystyle =\underset{h\to 0}{lim}-3\text{h}}$

= - 3(0) = 0   ....(1)

${\displaystyle \text{R}{\left[\text{f}\left(x\right)\right]}_{x=0}=\underset{x\to 0^{+}}{lim}\text{f}\left(x\right)=\underset{h\to 0^{+}}{lim}\text{f}(0+\text{h})}$

= ${\displaystyle \underset{h\to 0}{lim}\text{f}\left(\text{h}\right)=\underset{h\to 0}{lim}2\text{h}-\left|\text{h}\right|}$

= ${\displaystyle \underset{h\to 0}{lim}2\text{h}-\text{h}}$

= ${\displaystyle \underset{h\to 0}{lim}\text{h}=0}$   ...(2)

Also f(0) = 2(0) - |0| = 0

From (1), (2) and (3),

${\displaystyle \text{L}{\left[\text{f}\left(x\right)\right]}_{x=0}=\text{R}{\left[\text{f}\left(x\right)\right]}_{x=0}}$ = f(0) = 0

∴ f(x) = 2x - |x| is continuous at x = 0.