
“real: chapter_05” — 2011/5/22 — 22:55 — page 18 — #18
5-18 Real Analysis
2. Take f (x) as the characteristic function of the closed inter-
val [0, 1]. One can easily verify that f (0+) = 1, f (0−) =
0, f (1+) = 0, f (1−) = 1 so that 0 and 1 are points of dis-
continuities of the first kind. On the other hand, every point in
(0, 1) or (1, ∞) or (−∞,0) is a point of continuity for f .
3. f (x) =
1
x
for x > 0
0 for x ≤ 0
Here f :
R → R has discontinuities of the second kind at x = 0
since f (0−) = 0, f (0+) does not exist.
4. f (x) =[x], the greatest integer less than or equal to x. This
function f :
R → R is continuous at all points other than integers
and