Problem 59 - Calculus Early Transcendentals

Question

Suppose \(f\) is continuous at \(a\) and assume \(f(a)>0 .\) Show that there is a positive number \(\delta>0\) for which \(f(x)>0\) for all values of \(x\) in \((a-\delta, a+\delta) .\) (In other words, \(f\) is positive for all values of \(x\) in the domain of \(f\) and in some interval containing \(a .)\)

Step-by-Step Solution

Answer

Question: Prove that if a function f is continuous at point a, and f(a) > 0, there exists a positive number δ > 0 such that f(x) > 0 for all values of x in the open interval (a-δ, a+δ).

Step 1: Recall the definition of continuity

Since the function \(f\) is continuous at point \(a\), for any given value \(\epsilon > 0\), there exists a positive value \(\delta > 0\) such that \(|f(x) - f(a)| < \epsilon\) whenever \(|x - a| < \delta\).

Step 2: Choose an appropriate value for ε

Since we're given that \(f(a) > 0\), we need to find an interval where \(f(x) > 0\) as well. To guarantee this, we can choose the value of \(\epsilon\) to be equal to \(\frac{f(a)}{2}\). This means, we want to find a \(\delta\) such that $$|f(x) - f(a)| < \frac{f(a)}{2}$$ whenever \(|x - a| < \delta\).

Step 3: Show that \(f(x) > 0\) for \(|x - a| < \delta\)

We know that \(|f(x) - f(a)| < \frac{f(a)}{2}\). We can now manipulate this inequality to show that \(f(x) > 0\). Consider the following: $$|f(x) - f(a)| < \frac{f(a)}{2} \\ \Rightarrow -\frac{f(a)}{2} < f(x) - f(a) < \frac{f(a)}{2} \\ \Rightarrow f(a) - \frac{f(a)}{2} < f(x) < f(a) + \frac{f(a)}{2}$$ Now, since \(f(a) > 0\), it follows that \(f(a) - \frac{f(a)}{2} > 0\). Thus, we have shown that for the chosen value of \(ε=\frac{f(a)}{2}\): $$f(a) - \frac{f(a)}{2} < f(x)$$ whenever \(|x - a| < \delta\).

Step 4: Conclusion

We have proved that, given the continuity of \(f\) at \(a\) and \(f(a) > 0\), there exists a positive value \(\delta > 0\) such that \(f(x) > 0\) for all values of \(x\) in the open interval \((a-\delta, a+\delta)\).

Key Concepts

Continuous Functions

Understanding continuous functions is a fundamental part of calculus which directly ties into the concept of limits. In essence, a function is considered continuous at a point if there is no interruption in the graph of the function at that point. This means that the function’s value at a point is exactly where we expect it to be, given the function's values close to that point.

To put it in a more relatable way, think of driving smoothly along a road without hitting any potholes or bumps—that’s akin to a function's behavior being continuous at a location. A function is said to be continuous over an interval if it is continuous at every point within that interval. For example, the function given in the exercise is continuous around the point 'a', which implies that as we move along the graph of the function, we can expect no sudden jumps or gaps—at 'a' and around it, the journey is smooth.

Epsilon-Delta Definition of Continuity

To delve deeper into what it means for a function to be continuous at a point, mathematicians employ the 'epsilon-delta' definition. This formal definition relies on two arbitrary positive numbers, \( \epsilon \) and \( \delta \), which are used to measure distances along the y-axis and x-axis, respectively.

Here's how it works: If you can find a \( \delta \) that ensures \( |f(x) - f(a)| < \epsilon \) whenever \( |x - a| < \delta \) for any \( \epsilon > 0 \) you choose, then the function f is continuous at point a. It's like a mathematical guarantee that for every tiny wiggle (\( \epsilon \) small) in vertical height, you can find a sufficiently small wiggle (\( \delta \) small) in the horizontal distance, keeping the function values close to \( f(a) \) and ensuring no breaks in the function. The exercise provided demonstrates this principle by finding a \( \delta \) that maintains the function's positivity in a specific interval around 'a'.

Intervals and Inequalities

Intervals and inequalities are yet another crucial component of calculus, frequently used to express domains and ranges of functions, or in context, such as finding where a function stays positive or negative.

An interval is simply a way to describe a segment of the number line. For instance, the open interval \( (a-\delta, a+\delta) \) describes all the numbers between \( a-\delta \) and \( a+\delta \) not including the endpoints. To establish the continuity of a function and its behavior, we often use intervals to specify where we're zooming in.

Inequalities play a significant role in describing these intervals, serving as mathematical statements about the relative size or order of two objects. They're the key to showing how to control the outputs of a function: by knowing how to maneuver the inputs within a certain range. In the provided exercise, inequalities are used to prove that the function remains positive within the interval around the point 'a', further emphasizing the importance of this concept in understanding function behavior.