TIFR-2015-Maths-A-7

Question

Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct?

• A. If $g$ is continuous, then $f ∘ g$ is continuous
• B. If $f$ is continuous, then $f ∘ g$ is continuous
• C. If $f$ and $f ∘ g$ are continuous, then $g$ is continuous
• D. If $g$ and $f ∘ g$ are continuous, then $f$ is continuous

Answer 1

I suppose 4 one is correct as
for FoG to continuous we must have G must be continuous at point 'a' then F must be continuous at point G(a).