How do you use the definition of continuity to determine weather f is continuous at $f(x)= x-4$ if $x<=0$ and $x^2+x-4$ if x>0?

Question

The way in which the function $f$ is defined, it seems that its continuity is to be discussed at the pt. $x=0$ .

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Answer 1 · 2016-08-19T16:50:30

$f$ is continuous at $x=0$ , as discussed in the Explanation Section below.

The function $f$ is continuous at $x=0 iff lim_(xrarr0) f(x)=f(0)$ .

As $xrarr0$ from RHS, i.e., $x rarr0+, x>0, so, f(x)=x^2+x-4$ , a quadratic poly., known to be cont. on $RR$

$:. lim_(xrarr0+) f(x)=lim_(xrarr0+) x^2+x-4=-4....(1)$ .

As $xrarr0-, x<0, so, f(x)=x-4,$ a linear poly., known to be cont. on $RR$ .

$:. lim_(xrarr0-) f(x)=lim_(xrarr0-) x-4=-4..............(2)$ .

Also, $f(0)=[x-4]_(x=0)=-4................(3)$

From $(1)-(3), lim_(xrarr0) f(x)=f(0)$ .

Hence, $f$ is continuous at $x=0$