Welcome to this post on a topic in calculus.
Here, I want to discuss functions on \(\mathbb{R}^n\) and it is an extension to \(\mathbb{R}\)eal analysis.
In essence generalizing one calculus of one variable to more is the this topic.
Some funny overlap between Multivariable calculus and Linear algebra is that they both work with functions/maps \(f:\mathbb{R}^n \rightarrow \mathbb{R}^m\)
This set of definitions will provide us with means to later understand:
We first need to revisit the continuity definition and derivatives. In the Real Analysis the \(\epsilon - \delta\) notation has been used.
Suppose \(f:\mathbb{R} \rightarrow \mathbb{R}\) is a function defined on the real line, and there are two real numbers p and L.
One would say that the limit of \(f(x)\), as \(x\) approaches p, is L and written \[\lim_{x \to p} f(x) = L\] or alternatively, say \(f(x)\) tends to L as x tends to p, and written: \[f(x) \to L \text{ as } x \to p,\]
If the following property holds:
for every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |x − p| < δ implies |f(x) − L| < ε.
Symbolically, \[ \forall \varepsilon > 0, \quad \exists \delta > 0, \quad \forall x \in \mathbb{R} \, \] \[0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon \]
As show with tha animation, it is crucial to have the means to measure distances. It is required to have a general idea directions. Look Here
Let \( f \) be a continuous real-valued function on a closed interval \([a, b]\). Then, \( f \) attains both its maximum and minimum on \([a, b]\).
If \( f \) is differentiable on \([a, b]\) and \( f'(x) = 0 \) for all \( x \in [a, b] \), then \( f \) is constant on \([a, b]\).
Recall that in previous lessons, we explored the concept of continuous functions and their behavior on closed intervals.