- #1
math8
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Let F(x) = Integral from a to x of f dt (a belongs in [a,b])
How do we show that F(x) is continuous? (f is Lebesgue integrable on [a,b] )
How do we show that F(x) is continuous? (f is Lebesgue integrable on [a,b] )
math8 said:quasar987, thanks but I don't see how the DCT shows that F(x+h)-F(x)-->0. I know that
lim integral =integal lim
but does lim K(x+h,x)*f(t) tends to 0? If yes, why?
The function f is continuous on [a,b] if for every ε > 0, there exists δ > 0 such that for all x,y in [a,b], if |x-y| < δ, then |f(x)-f(y)| < ε.
In Riemann integrable functions, continuity means that the limit of the function as it approaches a certain point is equal to the value of the function at that point. In Lebesgue integrable functions, continuity means that the function is almost continuous, meaning that it is continuous except for a set of measure 0.
Yes, a function can be Lebesgue integrable but not continuous. A function can be discontinuous at a finite number of points and still be Lebesgue integrable.
To prove continuity of a function f on [a,b], one must show that for every ε > 0, there exists δ > 0 such that for all x,y in [a,b], if |x-y| < δ, then |f(x)-f(y)| < ε. This can be done by using the definition of continuity and using the properties of Lebesgue integrable functions.
Lebesgue integrable functions and continuity have many real-life applications in fields such as physics, engineering, economics, and finance. In physics, Lebesgue integrable functions are used to model physical phenomena, such as the movement of particles. In engineering, they are used to analyze and design systems. In economics and finance, they are used to model and predict market trends. Continuity is also important in real-life applications because it ensures that there are no sudden changes or disruptions in a system, making it more stable and predictable.