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CSIR NET Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free to read right now. Pick your subject below.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

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Tagged “integration”

“Riemann integrable ⇒ continuous almost everywhere fails for… (i.e. 'bounded ⇒ integrable')”
“|f| Riemann integrable ⇒ f Riemann integrable”

False — f = 1 on Q∩[0,1],−1\mathbb{Q} \cap [0,1], -1 elsewhere

“A composition of Riemann integrable functions is Riemann integrable”
“fn→0f_{n} \to 0 pointwise on [0,1]⇒∫01fn→0[0,1] \Rightarrow \int_{0}^{1} f_{n} \to 0”
“If ∫0∞f\int_{0}^\infty f converges then f(x) → 0”

False — f with a spike of height n and width 2/n32/n^{3} at each integer n

“Pointwise convergence implies convergence of the integrals”

False — fn=n⋅1(0,1/n)f_{n} = n\cdot1_{(0,1/n)} on [0,1]