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

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Answer 1 · 2017-02-09T21:56:57

see below

#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)#