Introductory university-level calculus, linear algebra, abstract algebra, probability, statistics, and stochastic processes.
Multivariate functions: limits and continuity
Get link
Facebook
X
Pinterest
Email
Other Apps
-
We now consider the functions of the form \[y = f(x_1, x_2, \cdots, x_n),\] that is, functions multiple independent variables \(x_1, x_2, \cdots, x_n\):
\[f: \mathbb{R}^n \longrightarrow \mathbb{R}.\]
An \(n\)-variable function like \(f(x_1, x_2, \cdots, x_n)\) has a subset of \(\mathbb{R}^n =(\mathbb{R}\times\mathbb{R}\times\cdots\times\mathbb{R})\) as its domain. In particular, we study the meaning of limits and continuity in high-dimensional spaces.
Remark. More generally, we may consider a function with its codomain in \(\mathbb{R}^m\). □
In the following, we mainly work with functions on \(\mathbb{R}^2\) rather than more general \(\mathbb{R}^n\). There is a huge gap between functions on \(\mathbb{R}\) and those on \(\mathbb{R}^2\). Compared to this gap, the conceptual difference between functions on \(\mathbb{R}^2\) and those on \(\mathbb{R}^n, n > 2\) is relatively small. Therefore, for most cases, understanding functions on \(\mathbb{R}^2\) is (mostly) sufficient.
The above figure shows the graph of \(f(x,y) = x^2 + y^2\). Generally, the graph of a two-variable function is a surface, as in this example.
We often apply the following procedure to understand the graph of a two-variable (or bivariate) function. First, fix the value of \(x\) to a constant, say, \(x = a\). Then consider the cross-section of \(x = a\) (a plane) and \(z = f(x,y)\), which is represented as \(z = f(a, y)\). Now, \(z = f(a, y)\) is a function of only \(y\), which can be plotted on the \(y\)-\(z\) plane. For \(f(x,y) = x^2 + y^2\), we have \(f(a,y) = a^2 + y^2\) which is a parabola. Then, we change the value of \(a\) and repeat the same process. We can apply the same procedure by fixing the value of \(y\) instead. Each curve in the above figure can be obtained by this procedure, and the result is the mesh representation of the surface.
Limit of multivariate functions
Let \(f(x,y)\) be a function on \(S \subset \mathbb{R}^2\), and \((a,b)\in \mathbb{R}^2\) a point. Let's consider the notion of limit of \(f(x,y)\) as the point \((x, y)\) approaches \((a,b)\).
To begin with, what do we mean by "\((x,y)\) approaches \((a,b)\)"? Clearly, there are infinitely many ways \((x,y)\) can approach \((a,b)\): along a straight line, in a spiral, in an arbitrary curve, etc. When we consider the limit of \(f(x,y)\), we need to consider all possible ways of approaching it.
Intuitively, the limit of \(f(x,y)\) as \((x,y)\) approaches \((a,b)\) can be defined as follows:
If \(f(x,y)\) approaches a constant value \(\alpha \in \mathbb{R}\) as \((x,y)\) approaches \((a,b)\), irrespectively of the way the approach is made, then we say \(f(x,y)\) converges to \(\alpha\) as \((x,y)\) approaches \((a,b)\), and denote \[\lim_{(x,y) \to (a,b)}f(x,y) = \alpha\] or \[f(x,y) \to \alpha \text{ as } (x,y) \to (a,b).\]
More formally, the definition is given as:
Definition (Limit of a function)
The function \(f(x,y)\) on \(S \subset \mathbb{R}^2\) is said to converge to \(\alpha\) as \((x,y) \to (a,b)\) if the following condition holds:
For all \(\varepsilon > 0\), there exists a \(\delta > 0\) such that, for all \((x,y) \in S\), if \((x,y) \in N_{\delta}(a,b)\) and \((x,y) \neq (a,b)\), then \(|f(x,y) - \alpha| < \varepsilon\).
Let \(f(x,y)\) and \(g(x,y)\) be functions such that
\[\begin{eqnarray}
\lim_{(x,y)\to(a,b)}f(x,y) &=& \alpha,\\
\lim_{(x,y)\to(a,b)}g(x,y) &=& \beta.
\end{eqnarray}\] Then, the following hold:
For any \(k, l\in\mathbb{R}\), \[\lim_{(x,y) \to (a,b)}\{kf(x,y) + lg(x,y)\} = k\alpha + l\beta.\]
If \(\beta \neq 0\), \[\lim_{(x,y) \to (a,b)}\frac{f(x,y)}{g(x,y)} = \frac{\alpha}{\beta}.\]
Proof. Exercise (similar to the one-variable version). ■
Example (Eg:polynomial). Clearly,
\[\begin{eqnarray}
\lim_{(x,y)\to (a,b)}x &=& a,\\
\lim_{(x,y)\to (a,b)}y &=& b.
\end{eqnarray}\]
Let \(f(x,y)\) be an arbitrary polynomial of \(x\) and \(y\). Then, \[\lim_{(x,y)\to (a,b)}f(x,y) = f(a,b).\] For example, if \(f(x,y) = x^3 + 2xy^2\), then \[\lim_{(x,y) \to (a,b)}f(x,y) = a^3 + 2ab^2.\]
For the rational function \(\frac{f(x,y)}{g(x,y)}\) of \(x\) and \(y\) where \(f(x,y)\) and \(g(x,y)\) are polynomials of \(x\) and \(y\), if \(g(a,b) \neq 0\), we have
Example. The function \(f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}\) does not have a limit as \((x,y) \to (0,0)\). To see this, consider a line \(l: y = mx\) and move \((x,y)\) towards \((0,0)\) along this line. Then, if \(x\neq 0\),
which converges to \(\frac{1-m^2}{1+m^2}\) as \(x \to 0\). However, this limit depends on the slope \(m\). For example, if \(m = 1\), the limit is 0; if \(m = 0\), the limit is 1. Thus, the "limit" of \(f(x,y)\) depends on the way how \((x,y)\) approaches \((0,0)\). Hence, \(f(x,y)\) has no limit at \((0,0)\). □
Example (Eg 1). Let us find the limit of \(f(x,y) = \frac{xy^2}{x^2 + y^2}\) as \((x,y) \to (0,0)\). Using the polar form \((x,y) = (r\cos\theta, r\sin\theta)\), we have
Let \(f(x,y)\) and \(g(x,y)\) be functions that are continuous at \((x,y) = (a,b)\). Then the following functions are also continuous at \((x,y) = (a,b)\):
\(kf(x,y) + lg(x,y)\) where \(k, l \in \mathbb{R}\) are arbitrary constants.
\(f(x,y)g(x,y)\).
\(\frac{f(x,y)}{g(x,y)}\) if \(g(a,b) \neq 0\).
Proof. Exercise (see the above Theorem (Properties of limits)). ■
Example. As we have seen on the above example (Eg:polynomial), for any polynomial function \(f(x,y)\) and any point \((a,b)\in \mathbb{R}^2\), we have
\[\lim_{(x,y)\to(a,b)}f(x,y) = f(a,b).\]
Therefore, all polynomial function are continuous everywhere in \(\mathbb{R}^2\). Also, any rational function \frac{f(x,y)}{g(x,y)} (with \(f, g\) being polynomial functions) is continuous at any \((a,b)\in\mathbb{R}^2\) provided that \(g(a,b) \neq 0\).
Example. Is the following function continuous on \(\mathbb{R}^2\)?
For \((x,y) \neq (0,0)\), \(f(x,y)\) is a rational function and its denominator is non-zero. Therefore, it is continuous on \(\mathbb{R}^2\setminus \{(0,0)\}\). In the above Example (Eg 1), we have seen that \(\lim_{(x,y)\to(0,0)\}f(x,y) = 0\). But \(f(0,0) = 0\) by the definition above. Therefore, \(f(x,y)\) is continuous at \((0,0)\). Hence, \(f(x,y)\) is continuous everywhere on \(\mathbb{R}^2\).
Finally, we provide the following theorems without proof.
Theorem (Intermediate Value Theorem)
Let \(f(x)\) be a continuous function on a path-connected set \(D\) in \(\mathbb{R}^n\) such that \(f(a)\neq f(b)\) for \(a, b \in D\). Then, for any value \(l\) between \(f(a)\) and \(f(b)\), there exists at least one \(c\in D\) such that \(f(c) = l\).
Exercise. Why should the domain \(D\) be path-connected?
Theorem (Extreme Value Theorem)
The continuous function \(f(x)\) on a bounded closed set \(F \subset \mathbb{R}^n\) has maximum and minimum values. That is, there exist \(c, d \in F\) such that \(f(c)\) and \(f(d)\) are the maximum and minimum values, respectively, of \(f(x)\).
Exercise. Why should the domain \(F\) be a bounded closed set?
Defining the birth process Consider a colony of bacteria that never dies. We study the following process known as the birth process , also known as the Yule process . The colony starts with \(n_0\) cells at time \(t = 0\). Assume that the probability that any individual cell divides in the time interval \((t, t + \delta t)\) is proportional to \(\delta t\) for small \(\delta t\). Further assume that each cell division is independent of others. Let \(\lambda\) be the birth rate. The probability of a cell division for a population of \(n\) cells during \(\delta t\) is \(\lambda n \delta t\). We assume that the probability that two or more births take place in the time interval \(\delta t\) is \(o(\delta t)\). That is, it can be ignored. Consequently, the probability that no cell divides during \(\delta t\) is \(1 - \lambda n \delta t - o(\delta t)\). Note that this process is an example of the Markov chain with states \({n_0}, {n_0 + 1}, {n_0 + 2}...
Generational growth Consider the following scenario (see the figure below): A single individual (cell, organism, etc.) produces \(j (= 0, 1, 2, \cdots)\) descendants with probability \(p_j\), independently of other individuals. The probability of this reproduction, \(\{p_j\}\), is known. That individual produces no further descendants after the first (if any) reproduction. These descendants each produce further descendants at the next subsequent time with the same probabilities. This process carries on, creating successive generations. Figure 1. An example of the branching process. Let \(X_n\) be the random variable representing the population size (number of individuals) of generation \(n\). In the above figure, we have \(X_0 = 1\), \(X_1=4\), \(X_2 = 7\), \(X_3=12\), \(X_4 = 9.\) We shall assume \(X_0 = 1\) as the initial condition. Ideally, our goal would be to find how the population size grows through generations, that is, to find the probability \(\Pr(X_n = k)\) for e...
The birth-death process Combining birth and death processes with birth and death rates \(\lambda\) and \(\mu\), respectively, we expect to have the following differential-difference equations for the birth-death process : \[\begin{eqnarray}\frac{{d}p_0(t)}{{d}t} &=& \mu p_1(t),\\\frac{{d}p_n(t)}{{d}t} &=& \lambda(n-1)p_{n-1}(t) - (\lambda + \mu)np_n(t) + \mu(n+1)p_{n+1}(t),~~(n \geq 1).\end{eqnarray}\] You should derive the above equations based on the following assumptions: Given a population with \(n\) individuals, the probability that an individual is born in the population during a short period \(\delta t\) is \(\lambda n \delta t + o(\delta t)\). Given a population with \(n\) individuals, the probability that an individual dies in the population is \(\mu n \delta t + o(\delta t)\). The probability that multiple individuals are born or die during \(\delta t\) is negligible. (The probability of one birth and one death during \(\delta t\) is also negligible.) Consequ...
Comments
Post a Comment