Problem 77 - Calculus

Question

Suppose that \(f\) is continuous on \([a, b]\). (a) Let \(G(x)=\int_{a}^{x} f(t) d t .\) Show that \(G\) is continuous on \([a, b]\) (b) Let \(F(x)\) be any antiderivative of \(f\) on \([a, b]\). Show that \(F\) is continuous on \([a, b]\).

Step-by-Step Solution

Answer

Both \( G(x) \) and \( F(x) \) are continuous on \([a, b]\).

Step 1: Understanding the integral function and its properties

Recognize that \( G(x) = \int_{a}^{x} f(t) \, dt \) represents the area under the curve \( f(t) \) from \( a \) to \( x \). This integral function is central in understanding continuity due to the fundamental theorem of calculus.

Step 2: Showing continuity of \( G(x) \)

According to the Fundamental Theorem of Calculus, if \( f \) is continuous on \([a, b]\), then \( G(x) \) is differentiable on \( (a, b) \) and continuous on \([a, b]\). Differentiability implies continuity, so \( G(x) \) is continuous on \([a, b]\).

Step 3: Antiderivatives and continuity

An antiderivative of a function \( f \), like \( F(x) \), satisfies \( F'(x) = f(x) \) for all \( x \) in the interval \([a, b]\). The derivative of \( F(x) \) exists and equals \( f \), which is continuous. Therefore, \( F(x) \) is differentiable and hence continuous on \([a, b]\).

Step 4: Conclusion of both results

Both \( G(x) \) and \( F(x) \) are continuous on \([a, b]\) due to the properties discussed above relating continuity, differentiability, and the Fundamental Theorem of Calculus.

Key Concepts

Continuity

Continuity is a fundamental concept in calculus, crucial for understanding how functions behave over intervals. A function is considered continuous on an interval if, informally speaking, you can draw it without lifting your pen off the paper. In more technical terms, a function \( f \) is continuous at a point \( x_0 \) if the limit of \( f(x) \) as \( x \) approaches \( x_0 \) is equal to \( f(x_0) \).
This concept translates beautifully when considering the integral function \( G(x) = \int_{a}^{x} f(t) \, dt \).

• Based on the Fundamental Theorem of Calculus, if \( f \) is continuous on \([a, b]\), then \( G(x) \) is continuous on \([a, b]\).
• This happens because the integral accumulates values smoothly over the interval.

Thanks to the nature of integrals and the continuous function \( f \), there are no interruptions, jumps, or gaps in \( G(x) \). Thus, understanding continuity in this context assures us that \( G(x) \) behaves predictably and smoothly, reflecting the uninterrupted input function \( f \).

Differentiability

Differentiability is closely tied to continuity. We can think of it as "one step further" than continuity. To say that a function is differentiable at a point means it not only behaves smoothly but also has a tangent line at that point. More technically, a function \( F(x) \) is differentiable at a point \( x_0 \) if the derivative \( F'(x_0) \) exists.
An important implication is that if a function is differentiable at \( x_0 \), it is also continuous there. Differentiability thus implies continuity, but not vice versa. For the antiderivative function \( F(x) \), which satisfies \( F'(x) = f(x) \), the existence of the derivative means:

• Since \( f(x) \) is continuous and \( F'(x) = f(x) \), \( F(x) \) is differentiable on \([a, b]\).
• This differentiability implies that \( F(x) \) is also continuous on \([a, b]\).

Thus, differentiability ensures that \( F \) doesn’t just "exist," it does so in a manner that is consistent across the entire interval \([a, b]\).

Antiderivative

An antiderivative is a function whose derivative returns the original function. In simpler terms, if \( F(x) \) is an antiderivative of \( f(x) \), then \( F'(x) = f(x) \). Understanding antiderivatives is crucial when learning about integrals and their properties.
When tackling a problem involving antiderivatives, such as showing \( F(x) \) is continuous over \([a, b]\), we rely on the continuity of the original function \( f \):

• If \( f \) is continuous, which it is by the problem statement, then the antiderivative \( F(x) \) will also be continuous.
• This continuity arises because any small change in \( x \) results in a small change in \( F \), a hallmark of continuous functions.

This interplay between derivatives and antiderivatives, stemming from functions like \( F(x) \), accentuates the balanced dance between integration and differentiation, confirming the continuity of antiderivatives when the original function is itself continuous.