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Question
Find all the points of discontinuity of f defined by f(x)=|x|−|x+1|
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Solution
f(x)=−x+(x+1)=1,x⩽−1=−x−(x+1)=−2x−1,−1<x⩽0=x−(x+1)=−1,o<xLHLatx=−1=1RHLatx=−1=−2(−1)−1=+1f(−1)=1asLHL=RHL=f(−1)andhencecontinuousLHLatx=0=−2(0)−1=−1RHLatx=0=−1f(0)=−2(0)−1=−1∴LHL=RHL=f(0)andhencecontinuoussof(x)iscontinuousisR.
f(x)=−x+(x+1)=1,x⩽−1=−x−(x+1)=−2x−1,−1<x⩽0=x−(x+1)=−1,o<xLHLatx=−1=1RHLatx=−1=−2(−1)−1=+1f(−1)=1asLHL=RHL=f(−1)andhencecontinuousLHLatx=0=−2(0)−1=−1RHLatx=0=−1f(0)=−2(0)−1=−1∴LHL=RHL=f(0)andhencecontinuoussof(x)iscontinuousisR.
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