Question

Discuss the continuity to the given by
$f\left(x\right)=\{\begin{array}{c}x,\text{if}x\ge 0\\ x^2,\text{if}x<0\end{array}$

Answer 1

$f\left(x\right)=\{xifx\ge 0xifx<0\}$

to check for continuity we find the left hand limit and right hand limit. If for a point 'a' in the domain $f\left(x\right)$

$\Rightarrow LHL=RHL=f\left(a\right)$

Then the function is continuous at $x=a$

for $x>0$

$f\left(x\right)=X,$

and we know that $f\left(x\right)=X$ ( identity function ) is a rational function well defined in the interval $(0,\infty )$ and is continuous,

For $x<0$

$f\left(x\right)=x^2$

this is a rational function in 'x' and hence is continuous in the above domain.

However the definition of the function changes at $x=0$ hence we need to check for continuity at $x=0$

LHL : ${lim}_{x\to 0^{-}}f\left(x\right)={lim}_{h\to 0}f(0-h)={lim}_{h\to 0}{(0-h)}^2={lim}_{h\to 0}{h}^2=0$

RHL :${lim}_{x\to 0^{+}}f\left(x\right)={lim}_{h\to 0}f(0+h)={lim}_{h\to 0}(0+h)={lim}_{h\to 0}h=0$

$\therefore LHL=RHL=f\left(0\right)$

Hence, the function is continuous at $x=0$

$\therefore$ the function is continuous $x\in R$ i.e, is continuous through out its domain.

Hence, solved.