Page 272 - DMTH401_REAL ANALYSIS

Real Analysis

Notes h b b
Þ ò - kdx £ ò f(x) dx £ ò k dx (why?)
a a
b
-
-
Þ - k(b a) £ ò f(x) dx £ k(b a)
a
b
-
-
 ò f(x)dx £ k(b a) = k b a
a
which completes the proof of the theorem.
Case (ii): a > b
In this case, interchanging a and b in the Case (i), you will get

a
-
ò f(x) dx £ k(a b)
b
b
i.e. ò - f(x) dx £ k(a b)
-
b
-
-
i.e. ò f(x) dx £ k(a b) = k b a .
Case (iii): a = b

In this case also, the result holds,
b
since f(x) dx = 0 for a = b and k b a = 0 for a = b.
-
ò
a
Let [a,b] be a fixed interval. Let R [a,b] denote the set of all Riemann integrable functions on this
interval. We have shown that if f,g C R [a,b], then f + g f.g and f for A Î R belong to R [a,b].
Combining these with Property, we can say that the set R [a,b] of Riemann integrable functions
is closed under addition, multiplication, scalar multiplication and the formatian of the absolute
value.

If we consider the integral as a function Int: R[a,b]  R defined by
b
Int (f) = f(x) dx
ò
a
with domain R [a,b] and range contained in R, then this function has the following properties:
lnt (f+g) = Int (f) + Int (g), lnt (f) = Int (f)
In other words, the function lnt preserves 'Vector sums' and the scalar products. In the language
of Linear Algebra, the function lnt acts as a linear transformation. This function also has an
additional interesting property such as

lnt (f) £ lnt(g)
whenever
f £ g .
We state yet another interesting property (without proof) which shows that the Riemann Integral
is additive on an interval.

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