Continuity And Differentiability

Question

Show that the function f defined by f(x) = |1–x+|x||, where x is any real number, is a continuous function.

Solution

Here f(x) = |1–x+|x||
Let g(x)=1–x+|x| and h(x) = |x|
∴ (h o g)(x) = h (g (x)) = h(1–x+|x|) = |1–x+|x||
Now polynomial function 1 – x is a continuous function.
Also |x| is a continuous function
We know that sum of two continuous function is a continuous function.
∴ 1–x+|x| is a continuous function.
Now (h o g)(x) = |1–x+|x || is the composite of two continuous functions h and g.
∴ f(x) = |1–x+|x|| is a continuous function.

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Continuity And Differentiability

Show that the function f defined by f(x) = |1–x+|x||, where x is any real number, is a continuous function.

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