CA Foundation · Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity
Which of the following functions is continuous for all real values of x?
f(x) = |x| is continuous for all real x. Its left and right limits at 0 both equal 0, which matches f(0). The other functions are undefined at some real point, so they are not continuous everywhere.
- Af(x) = 1/x
- Bf(x) = |x|Correct
- Cf(x) = (x² − 1)/(x − 1)
- Df(x) = √x
Explanation
|x| is continuous everywhere, including at x = 0, since both one-sided limits equal 0. 1/x is undefined at 0, (x² − 1)/(x − 1) is undefined at 1, and √x is not defined for negative x.
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