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#