Problem 25 Let $f$ be a real function con... [FREE SOLUTION]

Chapter 8: Problem 25

Let $f$ be a real function continuous at $a,$ and let $\left\langle a_{n}\right\rangle$ be a sequence converging to $a$. Prove that $$ \lim _{n \rightarrow \infty} f\left(a_{n}\right)=f(a). $$

Short Answer

Expert verified

The limit $\lim_{n \to \infty} f(a_n) = f(a)$ follows from the continuity of $f$ at $a$ and the convergence of $\langle a_n \rangle$ to $a$.

Step by step solution

Understand the definition of continuity

We are given that the function $f$ is continuous at the point $a$. By definition, a function $f$ is continuous at a point $a$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that whenever $|x - a| < \delta$, it follows that $|f(x) - f(a)| < \varepsilon$.

Recall the definition of sequence convergence

The sequence $\langle a_n \rangle$ is converging to $a$. This means that for every $\delta > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$, $|a_n - a| < \delta$.

Apply continuity to sequence limits

Because $f$ is continuous at $a$, and $\langle a_n \rangle$ converges to $a$, for every $\varepsilon > 0$, there exists a $\delta > 0$ so that if $|a_n - a| < \delta$ then $|f(a_n) - f(a)| < \varepsilon$.

Check sequence convergence condition

From convergence of $\langle a_n \rangle$ to $a$, for the $\delta$ given in Step 3, there exists $N \in \mathbb{N}$ such that for all $n \geq N$, $|a_n - a| < \delta$.

Prove the limit statement

For each $n \geq N$, $|a_n - a| < \delta$ implies $|f(a_n) - f(a)| < \varepsilon$. This means that $\lim_{n \to \infty} f(a_n) = f(a)$ since for every $\varepsilon > 0$, there exists $N$ such that for all $n \geq N$, $|f(a_n) - f(a)| < \varepsilon$. Thus, the sequence $f(a_n)$ converges to $f(a)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuity

Continuity is a fundamental concept in calculus and analysis. When we say a function is continuous at a point, it means there are no abrupt jumps or breaks at that point. Specifically, for a function $f$ to be continuous at a point $a$, it must satisfy the condition that for every positive number, $\varepsilon$, no matter how small, there is a corresponding positive number $\delta$. This $\delta$ ensures that whenever $|x - a| < \delta$, the difference $|f(x) - f(a)|$ is less than $\varepsilon$.
$\varepsilon$ and $\delta$ are like controls that measure closeness:

$\varepsilon$ represents how close $f(x)$ should be to $f(a)$.
$\delta$ determines how close $x$ should be to $a$.

In essence, continuity assures us that small changes in $x$ result in small changes in $f(x)$. This property is crucial in calculus because it allows us to apply various analytical techniques, such as calculating limits and derivatives, at particular points.

Real Function

Real functions are mathematical expressions that associate real numbers to real numbers. A function $f: \mathbb{R} \to \mathbb{R}$ means that for every input $x$ from the set of real numbers ($\mathbb{R}$), there is a specific output $f(x)$, which is also a real number. Understanding real functions involves:

Domain: The set of all possible input values ($x$-values).
Range: The set of all possible output values ($f(x)$-values).
Graphical Representation: Often depicted as a curve or line on the Cartesian coordinate system, showing the relationship between input and output values.

The concept of real functions is vital since they model a plethora of real-world phenomena and help in understanding other advanced topics in mathematics. They establish a relationship between variables and serve as the basis for more complex analysis.

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a certain value. Understanding limits is essential for grasping calculus concepts like differentiation and integration. Essentially, $\lim_{x \to a} f(x) = L$ means that as $x$ gets closer and closer to $a$, $f(x)$ gets closer and closer to $L$. Here are some key points regarding limits:

Limits help determine continuity at a point. If a function has a limit at $a$ and equals the function's value at $a$, the function is continuous at that point.
They provide insights into the function's behavior near important points, like maxima, minima, or points of inflection.

Limits can also be one-sided, meaning they consider approaching from either the left or the right. Utilizing limits with sequences, as in this exercise, involves considering the output of a function as the inputs from the sequence converge to a particular point. A deep understanding of limits assists in solving complex mathematical problems and is crucial for studying sequences and series.

91影视

Let \(f\) be a real function continuous at \(a,\) and let \(\left\langle a_{n}\right\rangle\) be a sequence converging to \(a\). Prove that $$ \lim _{n \rightarrow \infty} f\left(a_{n}\right)=f(a). $$

Short Answer

Step by step solution

Understand the definition of continuity

Recall the definition of sequence convergence

Apply continuity to sequence limits

Check sequence convergence condition

Prove the limit statement

Key Concepts

Continuity

Real Function

Limit of a Function

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