If f(x)=|x-1|+|x+3|, then discuss the continuity and differentiability of

Dottie Parra 2021-10-23 Answered
If f(x)=|x1|+|x+3|, then discuss the continuity and differentiability of the function at x=-3 and x=1.
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Expert Answer

au4gsf
Answered 2021-10-24 Author has 95 answers
Consider the given function
f(x)=|x1|+|x+3|
First, check the continuity of the function at x=1
Since a function f(x) is continuous at x=a if
limxaf(x)=limxa+f(x)=f(a)
Now check the continuity at the point x=1, then
limx1f(x)=limx1(|x1|+|x+3|)
=limx1(x1)+limx1(|x+3|)
=0+4
=4
And
limx1+f(x)=limx1+(|x1|+|x+3|)
=limx1+(x1)+limx1+(|x+3|)
=0+4
=4
Therefore, the function is continuous at x=1
Now, check the continuity at the point x=-3, then
limx3f(x)=limx3(|x1|+|x+3|)
=limx3(|x1|)limx3(x+3)
=(|31|)(3+3)
=|4|0
=4+0
=4
And
limx3+f(x)=limx3+(|x1|+|x+3|)
=limx3+|x1|+limx3+|x+3|
=(|31|)+(3+3)
=4+0
=4
And
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