Week 10 3-25 through 3-25, 2013

Included: Monotonicity. Linearity. Additivity. Bounded Constraint. Continuity of Integral Function for integrable functions. Continuous Functions are integrable. Fundamental Theorem (Derivative form) for Continuous Functions. FTofC (Evaluation form) for Continuous Functions. Mean Value Theorem for Integrals for Continuous Functions.

3-26. Additivity. Continuity of Integral Function for Integrable Functions. Continuous Functions are Integrable.

Excerpt from Spivak, Calculus, Ch 13

Theorem [Additivity]

Proof: [Converse proof omitted]

EOP

Theorem: Suppose `f ` is integrable on `[a,b]` and there exist m and M so that for all `x in [a,b]` `m \le f(x) \le M`. Then `m(b-a) \le \int_a^b f \le M(b-a).

Proof:

Now consider `F` on the interval `[c, c+h]`, of length `h>0`.

Then `F(c+h) = \int_a^{c+h} f ` and using additivity we have `F(c+h) - F(c) = \int_c^{c+h} f `.

By the previous result we have

`-Mh \le \int_c^{c+h}f
\le Mh` or

(1) `-Mh \le F(c+h) - F(c) \le Mh`

Now consider `F` on the interval `[c+h, c]`, where `h`<0, and length `-h>0`.

(1) `-Mh \le F(c+h) - F(c) \le Mh`

Now consider `F` on the interval `[c+h, c]`, where `h`<0, and length `-h>0`.