f[x
] := Floor[ 2
∗
x ];
Plot[ f[x],
{
x,
−
3
.
5, 3
.
5
}
, AspectRatio
−
>
Automatic, PlotRange
−
>
{ { −
3
.
5, 3
.
5
}
,
{−
4
.
5, 4
.
5
}}
];
Mathematica
will draw vertical lines connecting points that it shouldn’t, making the graph look like treads
and risers of a staircase, whereas only the treads are on the graph.
C01S02.054:
The function
f
is undefined at
x
= 1. The graph consists of the horizontal line
y
= 1 for
x >
1 together with the horizontal line
y
=
−
1 for
x <
1. There is a discontinuity at
x
= 1.
C01S02.055:
Given:
f
(
x
) = [[
x
]]. If
n
is an integer and
n
x < n
+ 1, then express
x
as
x
=
n
+ ((
x
))
where ((
x
)) =
x
−
[[
x
]] is the
fractional part
of
x
. Then
f
(
x
) =
n
−
x
=
n
−
[
n
+((
x
))] =
−
((
x
)). So
f
(
x
) is the
negative of the fractional part of
x
. So as
x
ranges from
n
up to (but not including)
n
+ 1,
f
(
x
) begins at 0
and drops linearly down not quite to
−
1. That is, on the interval (
n, n
+ 1), the graph of
f
is the straight
line segment connecting the two points (
n,
0) and (
n
+1
,
−
1) with the first of these points included and the
second excluded. There is a discontinuity at each integral value of
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 Spring '11
 CHENG
 Calculus, Vertex, Line segment

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