Discuss the continuity of the function `f(x)={{:(2x -1 "," x lt 1),(3x-2"," x ge 1):}`

Explanation

When `x lt 1`
f(x)= 2x-1 is a polynomial function
`therefore` f(x) is continuous in `x lt 1`
When `x gt 1`
f(x) =3x -2 is a polynominal function
`there` f(x) is continous in `x gt 1`
At x=1 L.H.L `=underset(xrarr1^-)lim f(x)=underset(hrarr0)limf(0-h)`
`=underset(xrarr0)lim2(1-h)-1=2-1=1`
R. H.L `=underset(xrarr1^+)lim f(x)=underset(hrarr0)limf(1+h)`
`=underset(h rarr0)lim 3(1+h) -2=3-2=1`
and f(1)=`3 xx1 -2 =1`
`therefore underset(x rarr 1^+)f(x) = f(1)=underset(x rarr 1^(-1))lim f(x)`
`therefore` f(x) is continuous at x=1
Therefore f(x) is a continuous function .