Problem 8 Find formulas for some functions... [FREE SOLUTION]

Chapter 2: Problem 8

Find formulas for some functions that are continuous on the intervals \((-\infty, 0)\) and \((0,+\infty),\) but are not continuous on the interval \((-\infty,+\infty)\).

Short Answer

Expert verified

The function \(f(x) = \frac{1}{x}\) is continuous on \((-\infty, 0)\) and \((0, +\infty)\) but not on \((-\infty, +\infty)\).

Step by step solution

Identifying Discontinuous Points

For a function to be continuous on all real numbers, it must not have any discontinuities within the interval ewline \((-\infty, \infty)\). Here, we need to find functions that specifically have discontinuities only at a particular point. A common point of discontinuity is at zero for some functions.

Choosing a Type of Discontinuous Function

Consider the function \(f(x) = \frac{1}{x}\). This function is continuous everywhere except at \(x = 0\) because division by zero is undefined. Therefore, this function is a candidate for being continuous on the intervals \((-\infty, 0)\) and \((0, +\infty)\) but not on \((-\infty, +\infty)\).

Verifying Continuity on Specified Intervals

Check the continuity of the chosen function \(f(x) = \frac{1}{x}\) on the intervals \((-\infty, 0)\) and \((0, +\infty)\). In these intervals, the function is continuous since x never equals zero and the function is well-defined. For any point \(x = a eq 0\) within these intervals, the function is both defined and the limit exists and equals \(f(a)\).

Confirming Discontinuity at the Problematic Point

At \(x = 0\), the function \(f(x) = \frac{1}{x}\) is not just discontinuous; it is undefined. The limit as \(x\) approaches zero does not exist because the left-hand limit and the right-hand limit diverge, confirming that the function is not continuous over \((-\infty, +\infty)\).

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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 that illustrates how a function behaves across an interval. To say a function is continuous at a certain point essentially means that you can draw the curve without lifting your pencil off the paper. Mathematically, a function \( f(x) \) is continuous at a point \( c \) if three conditions are met:

The function \( f(x) \) is defined at \( x = c \).
The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

For instance, consider the function \( f(x) = \frac{1}{x} \). While it is seamless across the intervals \((-\infty, 0)\) and \((0, +\infty)\), the point at \( x = 0 \) disrupts its continuity, as the function becomes undefined at this point.

Intervals

Intervals define the specific range within which a function is analyzed for continuity. They can be finite or infinite and can include open, closed, or mixed boundaries.

An interval like \((-\infty, 0)\) is known as a 'left-open interval', stretching from negative infinity to just before zero, but not including zero itself.
Similarly, \((0, +\infty)\) is a 'right-open interval', extending from just above zero to positive infinity, excluding zero.
To contrast, a closed interval like \([a, b]\) includes all numbers from \(a\) to \(b\), including both \(a\) and \(b\).

In our example, the function \( f(x) = \frac{1}{x} \) remains continuous on both \((-\infty, 0)\) and \((0, +\infty)\), but not on \((-\infty, +\infty)\) due to the discontinuity at zero.

Undefined Points

Undefined points in functions are those where a mathematical expression does not produce a valid result. For functions like \( f(x) = \frac{1}{x} \), the problem arises when the denominator is zero. Calculations such as division by zero are undefined, and thus, these points are crucial in determining where a function may be discontinuous.Often, undefined points are where you'll primarily encounter discontinuities. In our context, \( x = 0 \) is the only point where \( f(x) = \frac{1}{x} \) becomes undefined, breaking its continuity across the entire real number set.

Limit Analysis

Limit analysis helps us explore the behavior of functions as the input approaches a certain value, particularly near discontinuities or undefined points. For a continuous function at a point, the function's value and the limit as it approaches that point should coincide.For example, consider \( f(x) = \frac{1}{x} \) again. As \( x \) approaches 0 from either the positive or negative side:

The left-hand limit \( \lim_{x \to 0^-} \frac{1}{x} = -\infty \)
The right-hand limit \( \lim_{x \to 0^+} \frac{1}{x} = +\infty \)

Because these two limits do not agree, the overall limit does not exist, affirming the function's discontinuity at \( x = 0 \). Thus, limit analysis is essential in confirming a break or jump in continuity.

91影视

Find formulas for some functions that are continuous on the intervals \((-\infty, 0)\) and \((0,+\infty),\) but are not continuous on the interval \((-\infty,+\infty)\).

Short Answer

Step by step solution

Identifying Discontinuous Points

Choosing a Type of Discontinuous Function

Verifying Continuity on Specified Intervals

Confirming Discontinuity at the Problematic Point

Key Concepts

Continuity

Intervals

Undefined Points

Limit Analysis

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