Let r0 ∈ℝ and consider the function g0: ℝ \{0} →ℝ defined by g0(x):={ cos (1 / x) if x

step 1 For the problem given to us here, we want to first show that F is continuous, but not differenti

Let r0 ∈ℝ and consider the function g0: ℝ \{0} →ℝ defined by...

Let $r_{0} \in \mathbb{R}$ and consider the function $g_{0}: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by $$ g_{0}(x):=\left\{\begin{array}{ll} \cos (1 / x) & \text { if } x \neq 0 \\ r_{0} & \text { if } x=0 \end{array}\right. $$ Show that $g_{0}$ is not continuous at $0 .$ Define $G_{0}: \mathbb{R} \rightarrow \mathbb{R}$ by $G_{0}(x):=$ $\int_{0}^{x} \cos (1 / t) d t$. Show that $G_{0}$ is differentiable at 0 and $G_{0}^{\prime}(0)=0$, that is, $$ \lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} \cos \frac{1}{t} d t=0. $$

Let $r_{0} \in \mathbb{R}$ and consider the function $g_{0}: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by
$$
g_{0}(x):=\left\{\begin{array}{ll}
\cos (1 / x) & \text { if } x \neq 0 \\
r_{0} & \text { if } x=0
\end{array}\right.
$$
Show that $g_{0}$ is not continuous at $0 .$ Define $G_{0}: \mathbb{R} \rightarrow \mathbb{R}$ by $G_{0}(x):=$ $\int_{0}^{x} \cos (1 / t) d t$. Show that $G_{0}$ is differentiable at 0 and $G_{0}^{\prime}(0)=0$, that is,
$$
\lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} \cos \frac{1}{t} d t=0.
$$

Key Concepts

Squeeze Theorem

The Squeeze Theorem is employed to determine limits by ‘squeezing’ a function between two others with known limits. This theorem is particularly useful when dealing with oscillatory functions where direct evaluation of the limit is challenging, as it can often show that the limit is zero.

Limit Definition of the Derivative

The limit definition of the derivative involves computing the limit of the ratio of the change in the function value to the change in the independent variable as the change tends to zero. This approach is essential for proving differentiability especially when dealing with functions defined via integrals.

Oscillatory Behavior

Some functions exhibit rapid oscillations near a particular point, which prevent the establishment of a unique limit. This behavior is significant in analysis as it can lead to discontinuities, even when the oscillatory function is bounded.

Differentiability

Differentiability at a point requires the existence of a limit of the difference quotient as the increment approaches zero. A function can be differentiable at a point even if the integrand within its defining integral is highly oscillatory or even discontinuous at that point.

Continuity

Continuity in real analysis means that a function’s value at a point is the limit of its values as the variable approaches that point. If the limit of the function as the variable approaches a specific point does not equal the function’s value at that point, then the function is discontinuous there.

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