discuss the continuity of function f(x)= |x|+|x-1| in interval [-1,2]

It can be directly concluded by observation that the function will be continuous across R but further justification is given below. Note that both |x| and |x-1| are continuous across R though not differentiable at every value of x.

f(x) = |x| + |x-1|

It can be observe that the critical points for the modulus bound terms are 0 and 1. I.e. at x = 0, |x| changes its sign and at x = 1, |x-1| changes its sign.

For x Î[-1, 0]

f(x) = -x -(x-1) =-2x+1

This is a linear function and is continuous.

For x = 0

LHL = RHL = f(0) = 1

For x Î[0, 1]

f(x) = x -(x-1) =1

This is a constant linear function and is continuous.

For x = 1

LHL = RHL = f(1) = 1

For x Î[1, 2]

f(x) = x+(x-1) =2x-1

This is a constant linear function and is continuous.

Hence, f(x) is continuous in the interval [-1,2].

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