Question

Examine the continuity of the following:

|x + 2| + |x – 1|

Answer 1

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.