Discuss the continuity and differentiability of the function g(t) at t=1, where g(t)={t3,t≤1t,t&gt;1

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Discuss the continuity and differentiability of the function g(t) at t=1, where  g(t)={t3,t≤1t,t&amp;gt;1

Elizabeth Witte · Accepted Answer

Checking continuity at t=1  Given that:  g(t)={t3,t≤1t,t&amp;gt;1  Let us check the continuity at t=1 :  limt→1−g(t)=limt→1−(t3)=1  And,  limt→1+g(t)=limt→1+(t)=1.  Therefore, for t=1 g(1)=1 implies that: limt→1−g(t)=limt→1+g(t)=1  Therefore, the function g(t) is continue at t=1.   Checking differentiability  Let us check the differentiability at t=1 ; we will test whether LHD(at t=1) and RHD(at t=1) are equal or not  LHD at t=1:  limt→1−g(t)−g(1)t−1=limt→1−t3−1t−1=limt→1−(t2+t+1)=3.  RHD at t=1  limt→1−g(t)−g(1)t−1=limt→1−t−1t−1=1  Therefore, at t=1 LHD≠RHD.  So, The function is differentiable at t=1 and continuous .

Jeffrey Jordon · Accepted Answer

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Discuss the continuity and differentiability of the function g(t) at t=1, where

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