Question

Prove that if \(\mathbf{r}\) is a vector-valued function that is continuous at \(c,\) then \(\|\mathbf{r}\|\) is continuous at \(c\).

Short answer

The magnitude of a vector-value function \(|\mathbf{r}|\) is indeed continuous at any point \(c\) if the vector-value function \(\mathbf{r}\) is continuous at that point. This is established by applying the definition of continuity and the triangle inequality to \(\mathbf{r}\).

Step-by-step solution

Define Continuity

The definition of continuity at a point \(c\) is that a function \(f\) is continuous at \(c\) if (and only if) for every real number \(\varepsilon > 0\) there exists another real number \(\delta > 0\) such that for any real \(x\) within a \(\delta\) neighbourhood of \(c\) (but not equal to \(c\)), the value \(|f(x) - f(c)|\) is smaller than \(\varepsilon.\)

Apply the Definition of Continuity to r

The vector-valued function \(\mathbf{r}\) is given to be continuous at \(c.\) This means that for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that if \(| x - c | < \delta\), then \(|\mathbf{r}(x) - \mathbf{r}(c)| < \varepsilon.\)

Apply the Triangle Inequality

The triangle inequality for vector magnitude states that for any vectors \(\mathbf{a}\) and \(\mathbf{b}\), we have \(|\mathbf{a} - \mathbf{b}| \leq |\mathbf{a}| + |\mathbf{b}|\). If we substitute \(\mathbf{a} = \mathbf{r}(x)\) and \(\mathbf{b} = \mathbf{r}(c)\), and apply the triangle inequality backwards, by replacing \(\varepsilon\) with \(|\mathbf{r}(x)| - |\mathbf{r}(c)|\), we get |\mathbf{r}(x) - \mathbf{r}(c)| >= |\mathbf{r}(x)| - |\mathbf{r}(c)|. Since \(\mathbf{r}\) is continuous and the inequality holds, this implies that |\mathbf{r}(x)| - |\mathbf{r}(c)| < \varepsilon. Hence the function |\mathbf{r}| is continuous at \(c\).

Key concepts

Continuity Definition

When we talk about continuity, we're referring to a property of a function that allows it to have no sudden jumps or breaks at a given point. Specifically, a function is continuous at a point c if, intuitively speaking, you can draw the function at that point without lifting your pencil off the paper. Formally, the mathematical community has agreed upon a precise definition: for a function to be continuous at a particular point c, it must satisfy three conditions:

• The function is defined at c.
• The limit of the function as it approaches c exists.
• The limit of the function as it approaches c is equal to the function's value at c.

One way to picture this is to imagine following the graph of a function with your finger. If you can do so without any interruptions or jumps, then the function is continuous.There's a more technical way to express this concept too. A function f is continuous at a point c if for every positive number \(\varepsilon\), no matter how small, we can find another positive number \(\delta\) such that for every x within a \(\delta\)-neighborhood of c, the value |f(x) - f(c)| is smaller than \(\varepsilon\). This precise formulation underpins much of the work that mathematicians do when they study the behavior of functions at specific points.

Triangle Inequality

The triangle inequality is an important principle in the world of geometry and linear algebra. It helps us understand the relationship between the lengths of sides when dealing with triangles and, more abstractly, the magnitude of vectors. The inequality gets its name because, in any triangle, the length of one side must always be less than or equal to the sum of the lengths of the other two sides.
When translated into the language of vectors, which can be thought of as arrows pointing from one point to another, the triangle inequality states that for any two vectors \(\mathbf{a}\) and \(\mathbf{b}\), the magnitude of their difference is always less than or equal to the sum of their magnitudes. Mathematically, we write this as \(|\mathbf{a} - \mathbf{b}| \leq |\mathbf{a}| + |\mathbf{b}|\).
This principle is vital when connecting dots in vector space and surfacing proofs around vector functions and their continuity. When considering the continuity of a vector-valued function, the triangle inequality provides us with the necessary groundwork to compare vector magnitudes, making it easier to work through the more subtle aspects of the function's behavior.

Vector Magnitude

In the context of vectors, the term vector magnitude refers to the 'length' or 'size' of the vector. If you think of a vector as an arrow, the magnitude is essentially how long that arrow is. To calculate the magnitude of a vector \(\mathbf{v}\) with components \([v_1, v_2, ..., v_n]\) in n-dimensional space, you can use the formula \(\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}\).
This concept is similar to how you would determine the length of a line segment in a two-dimensional space using the Pythagorean theorem. Just as you can calculate the hypotenuse of a right-angled triangle, you can work out the magnitude of a vector. Understanding vector magnitude is crucial when discussing the continuity of vector-valued functions, as it relates directly to how the magnitude function \(\|\mathbf{r}\|\) behaves as we move around in the vector space.