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Step-by-Step Solution
Step 1: Understand the Given Function
We have
f(x) = x \cdot \left[\frac{x}{2}\right]
defined for
-10 < x < 10 ,
where
[t]
denotes the greatest integer less than or equal to
t .
Step 2: Identify Potential Points of Discontinuity
The greatest integer function
\left[\frac{x}{2}\right]
can change values only when
\frac{x}{2}
is an integer. Hence, discontinuities in
f(x)
can occur at values of
x
where
\frac{x}{2}
is an integer. Within the interval
(-10,10) ,
\frac{x}{2}
takes values in
(-5, 5) .
The integers in that range are
-4, -3, -2, -1, 0, 1, 2, 3,
and
4.
Thus, the corresponding
x
values where the greatest integer function might “jump” are
x = 2k
for
k \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}.
Step 3: Check Continuity at Each Candidate Point
• At
x = 0:
We see that
f(0) = 0 \cdot \left[\frac{0}{2}\right] = 0.
Just to the left and to the right of
0,
the function value also remains
0,
making
f(x)
continuous at
x=0.
• For
x = \pm 2, \pm 4, \pm 6, \pm 8:
These are points where
\frac{x}{2}
is a nonzero integer (namely
\pm1, \pm2, \pm3, \pm4 ),
so the greatest integer part
\left[\frac{x}{2}\right]
jumps at these points. Typically,
f(x)
will have a discontinuity there because
\left[\frac{x}{2}\right]
changes by 1.
Step 4: Count the Discontinuities
The potential discontinuities within
-10 < x < 10
occur at
x = \pm 2, \pm 4, \pm 6, \pm 8.
That is a total of
8
points. Checking more closely reveals that the function indeed has a jump at each of these points, confirming discontinuity there.
Step 5: Conclude the Number of Points of Discontinuity
Therefore, the number of points of discontinuity of the function
f(x) = x \left[\frac{x}{2}\right]
for
-10 < x < 10
is
8.
