Proposition: Differentiable Functions and Tangent-Linear Approximation

Let \(D\subseteq\mathbb R\) (\(D\) is a subset of real numbers) and let $a\in D$ be a point such that there is at least one real sequence $(x_n)_{n\in\mathbb N}$ convergent to $a$, i.e. $\lim_{n\to\infty}x_n=a.$

A function $f:D\to\mathbb R$ is differentiable at $a$ if and only if there is a constant $c\in\mathbb R$, such that $$f(x)=f(a)+c(x-a)+\phi(x),\quad x\in D,$$ where $\phi$ is a function for which $$\lim_{\substack{x\to a\\x\neq a}}\frac{\phi(x)}{x-a}=0.\quad\quad ( * )$$


In other words, the $f$ is differentiable at $a$ if and only if it is possible to draw in the graph of $f$ a linear function with the equation $$L(x)=f(a)+c(x-a).$$

The graph of $L$ is the tangent to the graph of $f$ at the point $a$.

Loosely speaking, by drawing the graph of $L$ (which is straight line) instead of drawing the graph of $f$ (which might be curved), in any neighborhood of $a$ we are likely to make an error, which can be expressed by the function $\phi$. The condition $( * )$ means that this error function is much smaller (tends faster to zero) as the difference $x-a$ does, as $x$ tends to the point $a.$

Proofs: 1 Corollaries: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983