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Question

The function f(x)=|x|+|x|x is

A
Discontinuous at origin because |x| is discontinuous there
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B
Continuous at origin
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C
Discontinuous at origin because both |x| and |x|x are discontinuous there
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D
Discontinuous at the origin because |x|x is discontinuous there.
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Solution

The correct option is B Discontinuous at the origin because |x|x is discontinuous there.
f(x)=|x|+|x|x

Left Hand Limit:
limx0|x|+|x|x=x1

Right Hand Limit:
limx0+|x|+|x|x=x+1

Given function f(x) is the combination of two functions.
If both functions are continuous at origin then the function f(x) will be continuous at origin.
But here |x| is continuous at origin, and |x|x is discontinuous at origin.
Therefore, f(x) will also be discontinuous at origin due to discontinuity of |x|x at origin.

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