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Question
Examine the continuity of the following:
|x + 2| + |x – 1|
Solution
Let f(x) = |x + 2| + |x – 1|
f(x) is defined for all points of R.
Let x0 be an arbitrary point in R.
Then `lim_(x -> x_0)f(x) = lim_(x -> x_0) (|x + 2| + |x - 1|)`
= `|x_0 + 2| + |x_0 - 1|` .......(1)
`f(x_0) = |x_0+ 2| + |x_0 - 1|` .......(2)
From equation (1) and (2) we get
`lim_(x -> x_0)f(x) = f(x_0)`
Thus the limit of the function f(x) exist at x = x0 and is equal to the value of the function at x = x0.
Since x = x0 is an arbitrary point in R, the above
result is true for all points in R.
Hence f(x) is continuous at all points of R.
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