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Draw the graph and discuss the continuit...

Draw the graph and discuss the continuity of `f(x)=[sinx+cosx],x in [0,2pi],` where [.] represents the greatest integer function.

Text Solution

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`f(x)=[sinx+cosx]=[g(x)]`
where `g(X)=sinx+cosx`
Let us first draw the graph of `y=g(x)`
`g(0)=1,g ((pi)/(4))=sqrt2,g((pi)/(2))=1`
`g((3pi)/(4))=0,g(pi)=-1,g((5pi)/(4))=-sqrt2`
`g((3pi)/(4))=-1,g((7pi)/(4))=0,g(2pi)=1`
Therefore, the graph of `y=g(x)` and hence the graph of `y=[g(x)]` are as shown in the following figure.
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