Flying_Goat said:A function defined on ℝ is continuous at x if given ε, there is a δ such that |f(x)-f(y)|<ε whenever |x-y|<δ. Does this imply that f(x+δ)-f(x)=ε? The definition only deals with open intervals so i am not sure about this. If this is not true could someone please show me a counter example for it?
Any help would be appreciated. Thanks.
A continuous function is a mathematical function that has no sudden jumps or breaks in its graph. This means that the function can be drawn without lifting the pen from the paper.
A function is continuous if it satisfies the three conditions of continuity: 1) the function is defined at the point, 2) the limit of the function at the point exists, and 3) the limit equals the value of the function at the point.
The limit of a function at a point is the value that the function approaches as the input (x) gets closer and closer to the given point. It is denoted as lim f(x) = L as x approaches a.
To prove that a function is continuous, you need to show that it satisfies the three conditions of continuity and that the limit of the function at every point is equal to the value of the function at that point. This can be done using the epsilon-delta definition of continuity or by using the properties of continuous functions.
The epsilon-delta definition of continuity states that a function f(x) is continuous at a point x=a if for every positive value of epsilon, there exists a positive value of delta such that |f(x)-f(a)| < epsilon whenever |x-a| < delta. In other words, the function's output can be made as close as desired to its input by restricting the input to be within a certain distance from the given point.