How do you use the definition of continuity to determine weather f is continuous at $2 - x$ $2-x$ if x<1, 1 if x=1 and $x^{2}$ $x^2$ if x>1?

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Answer 1 · 2017-02-09T21:56:57

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$f(x)={(2-x if x<1) , (1 if x=1), (x^2 if x>1)$

Here are the intervals $(-oo,1)(1,oo)$ . If we can show that

$lim_(x->1^- ) f(x)=f(1)$ and $lim_(x->1^+ ) f(x)=f(1)$ then we can prove that f(x) is continuous

$lim_(x->1^- ) f(x)=lim_(x->1^- ) 2-x=2-1=1$

$f(1)=1$

Since $f(1)=lim_(x->1^- ) 2-x$ f is therefore continuous from the left at 1.

$lim_(x->1^+ ) f(x)=lim_(x->1^+) x^2 =1^2 =1$

$f(1)=1$

Since $f(1)=lim_(x->1^+ ) x^2$ , f is therefore continuous from the right at 1. Hence f is continuous on $(-oo,oo)$

How do you use the definition of continuity to determine weather f is continuous at 2-x2−x2-x if x<1, 1 if x=1 and x^2x2x^2 if x>1?