relation between measurable function and continuous functions what is the relation between measurable function and continuous function? Which one implies the other? Examples for both type of functions. Whether sinx, cosx, tanx etx are measurable functions on real line?
 A: You cannot decide if a function is continuous or measurable just by a rule like $\sin x$ or $\cos x$. Besides the fact that you need to specify a domain and range for the functions, even that isn't enough specificity to talk about continuity and measurability.
To talk about continuity, the domain and range would have to be made into topological spaces by specifying the 'open' sets in each.
To talk about measurability, the domain and range would have to be made into measurable spaces by specifying the 'measurable' sets in each.
In principle, you could construct these to be independent of each other, in which case a function could be continuous and non-measurable or measurable and discontinuous.
In reality, we would usually specify a topology on the domain and would then make sure that the open sets are all measurable (the smallest such family of measurable sets being called the Borel $\sigma-$algebra of the topological space). 
This practice ensures that all continuous functions are measurable, which is probably the answer you're looking for.
