Thread: Prove that if f is integrable then kf is integrable.

Okay so this makes sense because if the integral of f exists then kf should exist if k is an element in the reals.

okay so there is a theorem in my book that says:
Let a,b, and k be real numbers. a< b. Let f and g be a real-valued functions that are Riemann integrable on [a,b]. Then integral b to a of kf equals k* integral b to a of f.

So i start out by saying e>0 and that there is a delta>0 such that ||P||<delta.

||P|| = max(xi - x(i-1))<delta

I just can't figure out how to incorporate a real number in when I have been solving proofs like Prove (f+g) = (f)+(g) when f and g are both functions and greater than or equal to zero. When i can solve by using inf and sup to prove ...... can i do that here as well?

Any help would be appreciated .......

I will continue looking at the problem

If f is integrable then $\displaystyle \int_a^b f(x) dx = \lim_{h \rightarrow 0} h[f(a+h) + f(a+2h) + \cdots + f(a+nh)] $ where $\displaystyle h = \frac{b-a}{n} $ and $\displaystyle a + nh = b $.

You also know that $\displaystyle \lim_{h \rightarrow 0} k h[f(a+h) + f(a+2h) + \cdots + f(a+nh)] = k \lim_{h \rightarrow 0} h[f(a+h) + f(a+2h) + \cdots + f(a+nh)] $ right?

Originally Posted by derek walcott

Okay so this makes sense because if the integral of f exists then kf should exist if k is an element in the reals.

okay so there is a theorem in my book that says:
Let a,b, and k be real numbers. a< b. Let f and g be a real-valued functions that are Riemann integrable on [a,b]. Then integral b to a of kf equals k* integral b to a of f.

So i start out by saying e>0 and that there is a delta>0 such that ||P||<delta.

||P|| = max(xi - x(i-1))<delta

I just can't figure out how to incorporate a real number in when I have been solving proofs like Prove (f+g) = (f)+(g) when f and g are both functions and greater than or equal to zero. When i can solve by using inf and sup to prove ...... can i do that here as well?

Any help would be appreciated .......

I will continue looking at the problem

I will note four things and you put them together.

1. $\displaystyle \sup_{x\in[x_{j-1},x_j]}\text{ }kf(x)=k\sup_{x\in[x_{j-1},x_j]}\text{ }f(x)$ (the same for infimums). Prove this.

2. From this we see that $\displaystyle U(P,kf)=\sum_{j=1}^{n}\left(\sup_{x\in[x_{j-1},x_j]}\text{ }kf(x)\right)\Delta x_j=$$\displaystyle \sum_{j=1}^{n}k\sup_{x\in[x_{j-1},x_j]}\text{ }f(x)\cdot \Delta x_j=k\sum_{j=1}^{n}\sup_{x\in[x_{j-1},x_j]}\text{ }f(x)\cdot \Delta x_j=k U(P,f)$

Similarly $\displaystyle L(P,kf)=kL(P,f)$

3. Using this we see that $\displaystyle U(P,kf)-L(P,kf)=k\left(U(P,f)-L(P,f)\right)$. (think the $\displaystyle \varepsilon$ definition of integrability).


This proves integrability.

Now, we know that $\displaystyle \int_a^b\text{ }kf=\sup_{P\in\mathcal{P}}\text{ }L(P,kf)=\sup_{P\in\mathcal{P}}\left(k\cdot L(P,f)\right)$.

Now apply the ideas of 1. to finish.