Let r0 ∈ℝ, and consider the function f0: ℝ →ℝ defined by f0(x)={ sin (1 / x) if x ≠ 0

step 1 We want to prove that this function is continuous at 0 .0. So for it to be continuous, we want t

Let r0 ∈ℝ, and consider the function f0: ℝ →ℝ defined by...

Let $r_{0} \in \mathbb{R}$, and consider the function $f_{0}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $$ f_{0}(x)=\left\{\begin{array}{ll} \sin (1 / x) & \text { if } x \neq 0 \\ r_{0} & \text { if } x=0 \end{array}\right. $$ (i) Show that $f_{0}$ is not continuous at 0 . Conclude that the function $x \mapsto$ $\sin (1 / x)$ for $x \in \mathbb{R} \backslash\{0\}$ cannot be extended to $\mathbb{R}$ as a continuous function. (ii) (ii) If $I$ is an interval and $I \subset \mathbb{R} \backslash\{0\}$, then show that $f_{0}$ has the IVP on $I$. If $I$ an interval such that $0 \in I$, then show that $f_{0}$ has the IVP on $I$ if and only if $\left|r_{0}\right| \leq 1$.

Let $r_{0} \in \mathbb{R}$, and consider the function $f_{0}: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
f_{0}(x)=\left\{\begin{array}{ll}
\sin (1 / x) & \text { if } x \neq 0 \\
r_{0} & \text { if } x=0
\end{array}\right.
$$
(i) Show that $f_{0}$ is not continuous at 0 . Conclude that the function $x \mapsto$ $\sin (1 / x)$ for $x \in \mathbb{R} \backslash\{0\}$ cannot be extended to $\mathbb{R}$ as a continuous function.
(ii) (ii) If $I$ is an interval and $I \subset \mathbb{R} \backslash\{0\}$, then show that $f_{0}$ has the IVP on $I$. If $I$ an interval such that $0 \in I$, then show that $f_{0}$ has the IVP on $I$ if and only if $\left|r_{0}\right| \leq 1$.

Key Concepts

Intermediate Value Property (IVP) and Darboux Property

The Intermediate Value Property states that if a function takes two values at two points within an interval, then it must take every value between these two values at some point within the same interval. This property, also known as the Darboux property, is essential in real analysis since it reflects the idea that a function changes its values in a 'continuous' manner, even if it is not entirely continuous everywhere.

Function Extension

Function extension involves defining a function at values where it was previously undefined, in such a way that certain properties, like continuity, are preserved. This concept is especially important when dealing with removable discontinuities, where a proper choice of extension can 'fill in' the gap and create a continuous function over a broader domain.

Continuity and Discontinuity

Continuity at a point in a function implies that the limit of the function as the variable approaches that point equals the function’s value at that point. Conversely, discontinuity occurs when this condition fails, often due to oscillations, jumps, or undefined behavior. This concept is fundamental in understanding the behavior of functions in calculus and analysis.

Limits

The concept of limits captures the behavior of functions as the input approaches a particular value, even if the function is not defined at that point. Limits help establish whether a function can be considered continuous at a point and are essential in analyzing the behavior of functions near problematic points, such as those causing discontinuities.

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