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Question

Show that the function f Defined by f(x)=|1-x+|x|| is a continuous  function 


Solution

f(x) = |1-x+|x||
Consider the function,
g(x)=1-x+|x| and h(x)=|x|,
Then..take composition
(h o g)(x)= h(g(x))
                = h(1-x+|x|)
                = |1-x+|x||
Now,
h(x)=|x| is continuous because modulus function is continuous.
g(x)= 1-x+|x|
Since (1-x) is polynomial it is continuous because polynomial functions are continuous, and |x| is continuous.
Since the sum of two continuous functions is continuous,
g(x)= 1-x+|x| is continuous. 
hence g(x) and f(x) both are continuous.So the composition of two continuous functions are also continuous.
So,
f(x)= |1-x+|x|| is continuous.

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