Chapter 3: Many Variables; Section 1: Differentiation; page 1
Function of Many Variables, Exercise 1
Question
For the following functions
find
the domain of definition
the regions of continuity and discontinuity
Reminder
Bounded sequence (the sandwich rule)
One can find in the literature many additional rules. Here is one very useful, which can be proven directly from the definition of convergence:
(1.1.3.14)
This rule works also for limits of zero and infinity.
A function is called continuous at point p0, if
(3.1.1.14)
For practical purposes, one iterates a limit by converging consecutively each one of the variables. If the limit depends on the order of the variables, the limit does not exist. However, if the limits are equal, there is no guarantee of the existence of the limit.
Another way for calculating the limit, is by approaching the point p0 from different directions. If the limit depends on the direction, the limit does not exist. A test of just a few particular directions, does not guarantee, the existence of the limit.
Parts 1-7
Solution for the function
The function is continuous for all the (x, y) values, with possible exceptions for xy=0, which should be investigated.
For x=0 and
,
the sine is undefined between −1 and +1, and therefore
As a consequence the function is discontinuous for the points of the y axis, with a possible exception of the origin x=y=0, which should be investigated separately.
From the symmetry between the variables, the previous statement holds also for the x axis.
For the origin we obtain
for any possible way of approaching the origin.
On the other hand the sine is undefined and bound between −1 and +1; in this approach. According to the sandwich rule,
we obtain that
independently of the approach toward the origin.
As a consequence the definition of
makes the function continuous at the origin.
Parts 8-10
Solution for the function
As in the case of the first function, this function is also continuous for all the (x, y) values, with possible exceptions for xy=0, which should be investigated.
As in the case of the first function, the use of the sandwich rule yields that the definition of
makes the function continuous at the origin.
Contrary to the previous function,
because of of the vanishing factor xy. Therefore the definition of the function as
makes the function continuous for all the x,y plane.
Score
By parts.
Any one of the 10 parts is worth 1 point.
By functions.
Correct solution for the first function alone, is worth 7 points.
Correct solution for the second function alone, is worth 6 points.
The two questions together are worth 10 points.