squaremeplease said:Homework Statement
i just don't know how to chose e in this case..
would it be |f(x) - f(y)| < e/3
any pointers are greatly appreciated
Continuity is a fundamental concept in real analysis that describes the smoothness or connectedness of a function. It means that the function has no abrupt changes or breaks in its graph and can be drawn without lifting the pencil from the paper.
In real analysis, a function f is continuous at a point a if the limit of f(x) as x approaches a exists and is equal to f(a). This can also be expressed using the delta-epsilon definition, which states that for any given epsilon, there exists a delta such that if the distance between x and a is less than delta, the distance between f(x) and f(a) will be less than epsilon.
Pointwise continuity means that a function is continuous at each individual point in its domain, while uniform continuity requires that the function remains continuous over the entire domain. In other words, for uniform continuity, the value of delta can be chosen independently of the point a, whereas for pointwise continuity it may depend on the specific point.
Yes, a function can be continuous without being differentiable. For example, the absolute value function is continuous at all points, but it is not differentiable at x = 0. This is because the derivative of the absolute value function is undefined at that point.
The intermediate value theorem states that if a continuous function takes on two different values at two points in its domain, then it must also take on all values in between those two points. In other words, a continuous function cannot "jump over" a value. This theorem is often used to prove the existence of solutions to equations that cannot be solved algebraically.