Let f :→be a continuous map, choose the correct statementa)f is boundedb)The image of f is an open subset of Rc)f(A) is bounded for all bounded subsets A of Rd)f-1(A) is compact for all compact subsets A of RCorrect answer is option 'D'. Can you explain this answer?

Question

Let f :&rarr;be a continuous map, choose the correct statementa)f is boundedb)The image of f is an open subset of Rc)f(A) is bounded for all bounded subsets A of Rd)f-1(A) is compact for all compact subsets A of RCorrect answer is option 'D'. Can you explain this answer?

Answer

Since f : ℝ→ℝ is a continuous map.

take f (x) = x, clearly 'f' is unbounded

∴ option (a) is incorrect

take f(x) = sin x

clearly f (R) = [-1,1] which is closed

∴ option (b) is incorrect

also take A = [-1.1] which is compact.

But f^-1 (A) = R, ⇒ f^-1(A) is not compact.

Therefore option (d) is incorrect.

Answer

Explanation:

f-1(A) is compact for all compact subsets A of R
- Since f is a continuous map, the preimage of a compact set under a continuous map is always compact.
- This property is a consequence of the fact that continuous functions preserve compactness.
- In other words, if A is a compact subset of Rd, then f-1(A) is also compact in R.
Therefore, the correct statement is that f-1(A) is compact for all compact subsets A of R.

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