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UPSC 2021 Maths Optional Paper 2 Q1c — Step-by-Step Solution

10 marks · Section A

Riemann integral · Real Analysis · asked 10× in 13 yrs · Read the full method →

Question

If $f$ is monotonic in $[a,b]$, prove $f$ is Riemann integrable in $[a,b]$.

Technique

Use equal partition; $U-L$ telescopes to $h\cdot[f(b)-f(a)]$, which $\to 0$.

Solution

Setup. WLOG $f$ monotonically increasing (similar argument for decreasing). $f$ is bounded: $f(a)\le f(x)\le f(b)$.

Criterion (Riemann). $f$ is Riemann integrable on $[a,b]$ iff for every $\epsilon>0$, there exists a partition $P$ with $U(P,f)-L(P,f)<\epsilon$.

Step 1 — Equal partition

Let $P_n$ be partition $a=x_0<x_1<\cdots<x_n=b$ with $x_i-x_{i-1}=(b-a)/n=h$.

Since $f$ is increasing, on $[x_{i-1},x_i]$:

• $\inf f=f(x_{i-1})$ (left endpoint).
• $\sup f=f(x_i)$ (right endpoint).

Step 2 — Compute $U-L$

$U(P_n,f)-L(P_n,f)=\sum_{i=1}^n[f(x_i)-f(x_{i-1})]\cdot h=h\sum_{i=1}^n[f(x_i)-f(x_{i-1})]$.

The sum telescopes: $\sum_i[f(x_i)-f(x_{i-1})]=f(x_n)-f(x_0)=f(b)-f(a)$.

So $U(P_n,f)-L(P_n,f)=h\cdot[f(b)-f(a)]=\dfrac{(b-a)[f(b)-f(a)]}{n}$.

Step 3 — Make $<\epsilon$

For given $\epsilon>0$, choose $n>\dfrac{(b-a)[f(b)-f(a)]}{\epsilon}$. Then $U-L<\epsilon$.

So $f$ satisfies the Riemann integrability criterion. $\blacksquare$

Answer