Let f:R→R  be a continuous and one-to-one function. Which of the following statements are necessarily true?

Correct option is C

Here f is continuous and one-one , this implies that there will be no break in map of f and no value repeats itself so it will be strictly monotonic , Now

Let f(x) = $$e^x$$​  this is both continuous and one one , now first derivative of f will be  f'(x) = $$e^x$$ which will always be positive , so f is strictly increasing

but if,

$$f(x)=e^{-x}$$ then  $$f'(x)=-e^{-x}$$ which will always be negative , that means in this case f is strictly decreasing ,

so from these examples we can say that f is either strictly increasing or strictly decreasing ,  also $$e^x$$ can never be 0 so f need not be onto .​