Let f:(a, b) →ℝ and c ∈(a, b) . Show that the following are...

Let $f:(a, b) \rightarrow \mathbb{R}$ and $c \in(a, b) .$ Show that the following are equivalent: (i) $f$ is differentiable at $c$. (ii) There exist $\alpha \in \mathbb{R}, \delta>0$ and a function $\epsilon_{1}:(-\delta, \delta) \rightarrow \mathbb{R}$ such that $f(c+h)=f(c)+\alpha h+h \epsilon_{1}(h)$ for all $h \in(-\delta, \delta)$ and $\lim _{h \rightarrow 0} \epsilon_{1}(h)=0$ (iii) There exists $\alpha \in \mathbb{R}$ such that $$ \lim _{h \rightarrow 0} \frac{|f(c+h)-f(c)-\alpha h|}{|h|}=0 $$ If the above conditions hold, then show that $f^{\prime}(c)=\alpha$.

Let $f:(a, b) \rightarrow \mathbb{R}$ and $c \in(a, b) .$ Show that the following are equivalent:
(i) $f$ is differentiable at $c$.
(ii) There exist $\alpha \in \mathbb{R}, \delta>0$ and a function $\epsilon_{1}:(-\delta, \delta) \rightarrow \mathbb{R}$ such that
$f(c+h)=f(c)+\alpha h+h \epsilon_{1}(h)$ for all $h \in(-\delta, \delta)$ and $\lim _{h \rightarrow 0} \epsilon_{1}(h)=0$
(iii) There exists $\alpha \in \mathbb{R}$ such that
$$
\lim _{h \rightarrow 0} \frac{|f(c+h)-f(c)-\alpha h|}{|h|}=0
$$
If the above conditions hold, then show that $f^{\prime}(c)=\alpha$.

Key Concepts

Differentiability

Differentiability refers to a function having a well-defined tangent at a point, meaning there exists a linear approximation that accurately reflects the function's behavior close to that point. This concept is typically formalized by demanding that the limit of the difference quotient exists as the increment approaches zero.

Limit Definition of the Derivative

The derivative at a point is fundamentally defined as the limit of the difference quotient, that is, the limit of [f(c+h) - f(c)]/h as h tends to zero. This definition forms the backbone of differential calculus and is central to understanding the instantaneous rate of change of the function.

Linear Approximation and Error Term

A key perspective on differentiability is expressing the function locally as a linear function plus an error term that vanishes in a specified sense. This is captured by writing f(c+h) = f(c) + ?h + h?(h), where the error function ?(h) tends to zero as h approaches zero, indicating that the linear part is the dominant behavior near the point.

Little-o Notation

Little-o notation provides a concise way to describe the behavior of the error term in relation to h. When we say that f(c+h) = f(c) + ?h + o(h), it means that the error term is insignificant compared to h as h becomes very small. This formalism is key in establishing precise equivalences between different formulations of differentiability.

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