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h(x) may or may not be continuous in [a, c]`h(b^(+))=g(b^(-)) and h(b^(-))=f(b^(+))``h(b^(-))=g(b^(+)) and h(b^(+))=f(b^(-))`h(x) has a removable discontinuity at x = b

Answer :

C::DSolution :

Given f is continuous is [a, b]`" (i)"` <br> g is continuous in [b, c]`" (ii)"` <br> `g(b)=g(b)" (iii)"` <br> Also, `h(x)={{:(f(x)",",x in[a,b)),(f(b)=g(b)",",x=b),(g(x)",",x in (b,c)):}` <br> From (i) and (ii), we can conclude that h(x) is sontinuous in <br> `[a,b)uu(b,c]`. <br> Also, `f(b^(-))=f(b),g(b^(+))=g(b)` <br> `therefore" "h(b^(-))=f(b^(-))=f(b)=g(b)=g(b^(+))=h(b^(+))` <br> Obviously, `g(b^(-)) and f(b^(+))` are undefined. <br> `h(b^(-))=f(b^(-))=f(b)=g(b)=g(b^(+))` <br> and `h(b^(+))=g(b^(+))=g(b)=f(b)=f(b^(-))` <br> Hence, `h(b^(-))=h(b^(+))=f(b)=g(b)` <br> Thus, h(x) has removable discontinuity at x = b.**Revision of limits**

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