f is a continous function in `[a, b]`; g is a continuous function in [b,c]. A function h(x) is defined as `h(x)=f(x) for x in [a,b) , g(x) for x in (b,c]` if f(b) =g(b) then

Options

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

Answer

C, D

Explanation

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