I really need help with checking this. I haven't yet seen such a problem.
The task is to compare the cardinality of set of all functions from R to R and cardinality of all continuous functions from R to R. There's a hint that every continuous function is well defined by its values on rational numbers.
Suppose we know all the values of function at rational numbers. Then between every two rational numbers lies at least one irrational. Since the function is continuous, its values at irrational numbers have to lie between its values at rational. Hence, the theorem holds.
Since cardinality of R is c (continuum) the first cardinal number is [MATH]c^c[/MATH], the second is [MATH]N^c[/MATH] because there's a bijection between Q and N. But then I'm a little lost in how to compare there two numbers. Any hints? Thanks !!!!!!!
The task is to compare the cardinality of set of all functions from R to R and cardinality of all continuous functions from R to R. There's a hint that every continuous function is well defined by its values on rational numbers.
Suppose we know all the values of function at rational numbers. Then between every two rational numbers lies at least one irrational. Since the function is continuous, its values at irrational numbers have to lie between its values at rational. Hence, the theorem holds.
Since cardinality of R is c (continuum) the first cardinal number is [MATH]c^c[/MATH], the second is [MATH]N^c[/MATH] because there's a bijection between Q and N. But then I'm a little lost in how to compare there two numbers. Any hints? Thanks !!!!!!!