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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 “uniform convergence”

“Pointwise limit of continuous functions is continuous”
“fn→ff_{n} \to f uniformly ⇒fn′→f′\Rightarrow f_{n}' \to f'”

False — fn(x)=sin⁡(f_{n}(x) = \sin(nx)/n

“fn→0f_{n} \to 0 pointwise on [0,1]⇒∫01fn→0[0,1] \Rightarrow \int_{0}^{1} f_{n} \to 0”
“A power series converges uniformly on its open disc of convergence”
“A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence”

False — fn(x)=xnf_{n}(x) = x^{n}