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  • Let f is a continuous function in [a, b], g is a continuous function in [b, c]. A function h(x) is defined as <img alt="\mathrm{h(x)=f(x) \text { for all } x \in[a, b)=g(x) \text { for } x \

Let f is a continuous function in [a, b], g is a continuous function in [b, c]. A function h(x) is defined as
\mathrm{h(x)=f(x) \text { for all } x \in[a, b)=g(x) \text { for } x \in(b, c] }
\mathrm{ \text { if } f(b)=g(b) \text { then : } }
 

Option: 1

\mathrm{h(x) \text { has a removable discontinuity at } x=b}


Option: 2

\mathrm{ h(x) \text { may or may not be continuous in }[a, c]}


Option: 3

\mathrm{ h\left(b^{-}\right)=g\left(b^{+}\right) \text {and } h\left(b^{+}\right)=f\left(b^{-}\right) }


Option: 4

\mathrm{ h\left(b^{+}\right)=g\left(b^{-}\right) \text {and } h\left(b^{-}\right)=f\left(b^{+}\right) }


Answers (1)

best_answer

Given, f(x) is continuous in [a, b]
g(x) is continuous in [b, c]
\mathrm{ \begin{aligned} & f(b)=g(b) \\ & h(x)=f(x) \text { for } x \in[a, b] \end{aligned} }
\mathrm{ g(x) \, for \, x \in(b, c) }
\mathrm{ h(x) is \, \, \, continuous \, \, in [a, b] \cup(b, c] using ..........eq. (i). }
\mathrm{ Also, f\left(b^{-}\right)=f(b) ; g\left(b^{+}\right)=g(b) }
\mathrm{ \therefore \quad h\left(b^{-}\right)=f\left(b^{-}\right)=f(b)=g(b)=g\left(b^{+}\right)=h\left(b^{+}\right) }
\mathrm{ Using eq. (iv) and eq. (v). g\left(b^{-}\right)and f\left(b^{+}\right)are undefined. }
\mathrm{ \begin{aligned} & h\left(b^{-}\right)=f\left(b^{-}\right)=f(b)=g(b)=g\left(b^{+}\right) \\ & h\left(b^{+}\right)=g\left(b^{+}\right)=g(b)=f(b)=f\left(b^{-}\right) \\ & h\left(b^{-}\right)=h\left(b^{+}\right)=f(b)=g(b) \end{aligned} }
\mathrm{ Thus, h(b) may or may not be continuous in [a, c]. }

 

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