Question

the value of x for which f(x)=|x²-1| is discontinuous

Answer 1

Mar 11, 2017 - Ex 5.1, 34 Find all the points of discontinuity of f defined by   =  – +1 . When ≥0   = − +1 −x − x + 1 = x − x − 1 .. ... is continuous for all real values . Hence   = −− is continuous at −≤< ⇒ There is no point of discontinuity Hence   is continuous for all ∈

Answer 2

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f(x)=[x2+1]

In the interval xϵ[1,3],

f(1)=[12+1]=2f(3)=[32+1]=10

Clearly, all possible values of f(x) in the interval [1,3] must be 2,3,4,5,6,7,8,9,10.

The greatest integer function will become discontinuous at the eight points where [x2+1]=2,3,4,5,6,7,8,9,10.

Explanation: If [x2+1]=n, this means

n≤x2+1<n+1⇒x2−(n−1)≥0andx2−n<0

For positive x, the solution is:

⇒xϵ(n−1,n)

The function becomes discontinuous every time the value of n changes. This value changes for n=2,3,4,5,6,7,8,9,10.

So, the function is continuous everywhere in the interval [1,3] except at nine points.

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