This was a very nice puzzle – not an easy one and as such I received answers from the classical die hards. People who sent the correct answers include Suman Saraf and his daughter (I think age 10!), Sid Mulherkar, Pratik Poddar and Vishal Poddar. Well done all.
Copying answers from Vishal as well as Suman that explain the same thing:
This is based on a mathematical theorem known as the Intermediate Value Theorem. Interested reader can read more at https://en.wikipedia.org/wiki/Intermediate_value_theorem.
Hope you enjoyed the puzzle!