Is there a continuous function from Ru onto Rc From RFC onto

Is there a continuous function from R_u onto R_c? From R_FC onto R_C?

Solution

Ans- A function f is continuous when, for every value c in its Domain: f(c) is defined, and: \"the limit of f(x) as x approaches c equals f(c)\" The limit says: \"as x gets closer and closer to c then f(x) gets closer and closer to f(c)\" And we have to check from both directions: as x approaches c (from left) then f(x) approaches f(c) AND as x approaches c (from right) then f(x) approaches f(c) If we get different values from left and right (a \"jump\"), then the limit does not exist! How to Use: Make sure that, for all x values: f(x) is defined and the limit at x equals f(x) Here are some examples: Example: f(x) = (x2-1)/(x-1) for all Real Numbers The function is undefined when x=1: (x2-1)/(x-1) = (12-1)/(1-1) = 0/0 So it is not a continuous function Let us change the domain: Example: g(x) = (x2-1)/(x-1) over the interval x<1 Almost the same function, but now it is over an interval that does not include x=1. So now it is a continuous function (does not include the \"hole\") Example: How about this piecewise function: which looks like: It is defined at x=1, because h(1)=2 (no \"hole\") But at x=1 you can\'t say what the limit is, because there are two competing answers: \"2\" from the left, and \"1\" from the right so in fact the limit does not exist at x=1 (there is a \"jump\") And so the function is not continuous.