Which of the following statements are true or false? Give proofs or counterexamples to justify your answers.I said that this statement was false and used the counterexample andi). If is continuous and is such that , then f is continuous.
I put true.ii). If and are both continuous and , then f=g.
Since g is continuous I know that .
f is also continuous so .
Define . WLOG, let . is this step valid?
This gives as required.
Since my initial definitions only apply to a rational c, would I need to prove they are equal when c is not rational?
I opted for true.iii). If is continuous at 0 with and is bounded then the product fg is continuous at 0.
f is continuous:
Since I am only interested in the behaviour of the multiple functions at 0, pick . can you do this? I need to restrict delta somehow so large values of delta can also be used.
This gives that:
I said this is true since the composition of two continuous functions is itself continuous. However, I cannot formulate a proof!If is such that the composition (this is a mapping, not a limit) is continuous on then f is continuous.
I have a feeling this is false. Unfortunately I am unable to think of a counterexample.If is such that the composition (mapping once again) is continuous on then f is continuous.
Any help with any of these questions would be appreciated. They get so much harder towards the end!