Prove that if 𝐫 is a vector-valued function that is continuous at c, then 𝐫 is continuous at c.

Name: Problem 85, Chapter 12: Calculus Early Transcendental Functions
Uploaded: 2021-09-29T22:20:25-08:00
Duration: 1 min 30 s
Channel: Monica Miller
Description: Prove that if 𝐫 is a vector-valued function that is continuous at c, then 𝐫 is continuous at c.

step 1 Prove that if R is a vector valued function that is continuous at C, then its magnitude is conti

Prove that if 𝐫 is a vector-valued function that is...

Key Concepts

Vector-Valued Function

A vector-valued function assigns a vector to each point in its domain. Unlike scalar functions that output a single number, vector-valued functions output vectors from ??. They are fundamental in multivariable calculus and physics to represent quantities that have both magnitude and direction.

Continuity of a Function

Continuity of a function at a point means that the limit of the function as the variable approaches that point equals the function’s value at that point. For vector-valued functions, continuity is defined component-wise, meaning that each component function must be continuous at the point in question.

Norm (Magnitude) of a Vector

The norm of a vector is a measure of its length or magnitude, typically defined in Euclidean space as the square root of the sum of the squares of its components. The norm is a scalar and provides a way to quantify the size of a vector.

Continuity of the Norm Function

The continuity of the norm function means that applying the norm to a vector-valued function preserves continuity. This relies on the fact that the norm is a continuous function itself, and when composed with a continuous vector-valued function, the result is a continuous scalar function.

Limit Properties and Composite Functions

When dealing with limits in analysis, if a function is continuous and is composed with another function that has a limit, the continuity of the outer function lets you pass the limit through it. This principle is key in proving that the norm of a vector-valued function is continuous because the norm function is continuous.

Transcript

Please give Ace some feedback