Function Arbitrary

Function Arbitrary

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Let X={1,2,3} and a function f:X-->R is such that f(1)=3, f(2)=4, f(3)=5. If f continuous at 2? see details

Explain the answer. Now let x be the set of all positive integers and Q: X--> R be some arbitrary function. Take an arbitrary positive integer n. Is Q continuous at n?
Explain the answer. Now let x be the set of all positive integers and Q: X--> R be some arbitrary function. Take an arbitrary positive integer n. Is Q continuous at n?

Guys, please HELP!!! ITS URGENT!!!

The answer to both these questions is yes. Informally speaking, a function is continuous at some point (x,y) if it sends values in the domain of the function which are close to x to values in the range of the function which are close to y. For both f and Q, all values in the domain are isolated, meaning that each value in the domain is separated by some positive distance from all other values in the domain. So, for each point (x,y) in the graph of the function, x is the only value in the domain which is close to x, and it is sent to y, which is close (in fact equal to) y.

More formally, the ε-δ definition of continuity states that f is continuous at a point a if for any ε > 0, there exists some δ > 0 such that for all x in the domain of f, if |x − a| < δ, then |f(x) − f(a)| < ε. This definition can be shown to hold by finding a suitable value of δ for each ε and showing that this value of δ satisfies the definition. If for every ε we pick δ = 1/2, then for any a ∈ X, x = a is the only element of X satisfying |x − a| < δ, and then f(x) = f(a) so |f(x) − f(a)| < ε. The proof for Q is very similar; we can again take δ = 1/2.

33220A Function / Arbitrary Waveform Generator, 20 MHz

Function Arbitrary