2Let f:[a, b] →ℝ be integrable. Define G:[a, b] →ℝ by G(x):=∫x^b f(t) d t . Show that G is conti

step 1 Okay, hello. So here we want to show that the absolute value of the integral going from A to B o

2Let f:[a, b] →ℝ be integrable. Define G:[a, b] →ℝ by...

2Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable. Define $G:[a, b] \rightarrow \mathbb{R}$ by $$ G(x):=\int_{x}^{b} f(t) d t . $$ Show that $G$ is continuous on $[a, b] .$ Further, show that if $f$ is continuous at $c \in[a, b]$, then $G$ is differentiable at $c$ and $G^{\prime}(c)=-f(c)$. (Hint: Propositions $6.7,6.20$, and $6.21$.)

2Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable. Define $G:[a, b] \rightarrow \mathbb{R}$ by
$$
G(x):=\int_{x}^{b} f(t) d t .
$$
Show that $G$ is continuous on $[a, b] .$ Further, show that if $f$ is continuous at $c \in[a, b]$, then $G$ is differentiable at $c$ and $G^{\prime}(c)=-f(c)$. (Hint:
Propositions $6.7,6.20$, and $6.21$.)

Key Concepts

Riemann Integrability

This concept refers to the condition under which a function on a closed interval has a well-defined integral, typically defined as the limit of Riemann sums. Riemann integrability is crucial because it provides the foundational framework for working with the definite integral and ensures that the function can be integrated over any subinterval, even if the function may exhibit discontinuities at some points.

Continuity of the Integral Function

The property that the function defined by an integral with a variable limit is continuous relies on the integrability of the original function. This result is derived from the fact that small changes in the limit of integration result in the addition of a small change in the integral value. This principle is a cornerstone in real analysis and allows one to conclude that the integral function is continuous over the interval.

Fundamental Theorem of Calculus

This theorem builds a bridge between integration and differentiation by stating that if a function is continuous on an interval, then the integral of the function with a variable upper or lower limit is differentiable, and its derivative is the original function (or its negative, depending on the integration limits). It provides the rigorous justification for differentiating an integral to retrieve the original integrand, and it is a central result in calculus.

Differentiability at a Point

Differentiability involves showing that the derivative of the function exists at a particular point. In the context of an integral function defined by variable limits, if the original function is continuous at a point, then one can use the limit definition of the derivative to show that the integral function is differentiable at that point. The argument typically exploits the uniform continuity of the integrable function on the interval and provides the exact expression for the derivative at the point in question.

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