prove that f(x)=1+x+|x| is continuous
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Show that f:R→R, f(x)=1/x is continuous at any c≠0.
Notice: (choose your δ so that you stay away from 0)
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clearly f(x) is defined for every real x as
f(x) = 1+x+x = 1+2x when x>= 0
1+x-x = 1when x < 0
hence function f(x) changes at x= 0
so if f(x) is continuous at x=0, it will be continuous for all real x
now f(0)from left = 1
f(0) from right = 1+2*0 = 1
so f(0) from left = f(0) from right
hence f(x) is continuous at x= 0
so f(x) is continuous for all real x
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