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Show that the function f defined by f(x...

Show that the function f defined by
`f(x)={(x if "x is rational"),(-x if "x is irrational"):}`
is continuous at x=0 `AAxne0inR`

Text Solution

Verified by Experts

f(0)=0
L.H.L=`lim_(xto0)f(x)=lim_(hto0)f(-h)`
`=lim_(hto0){(-h if "h is rational" =0),(h if "h is irrational"):}`
Similarly R.H.L.=0
Thus L.H.L.=R.H.L.=f(0)
Hence f(x) is continuous at x=0
We can easily show that f(x) is discontinuous at all real points `xne0`
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