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JG89
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If f is continuous at x = a, then is it continuous in some neighborhood of x = a as well?
lurflurf said:No
A standard example is
f(x)=x x rational
f(x)=0 x irrational
f is continuous at x=0
Continuity is a concept in mathematics that describes the behavior of a function as it approaches a certain point. A function is considered continuous if there are no sudden jumps or breaks in its graph.
A function is considered continuous if the limit of the function at a point exists and is equal to the value of the function at that point. This means that the function approaches the same value from both the left and right sides of the point.
Continuity is an important concept in mathematics because it helps us understand the behavior of functions and allows us to make predictions and solve problems related to them. It is also a fundamental concept in calculus and is used to define derivatives and integrals.
The three types of continuity are point continuity, uniform continuity, and differentiability. Point continuity means that a function is continuous at a specific point, while uniform continuity means that a function is continuous over a certain interval. Differentiability means that a function is continuous and has a well-defined derivative at a point.
Continuity is used in a wide range of real-world applications, including physics, engineering, economics, and biology. For example, in physics, continuity is used to describe the motion of objects, while in economics, it is used to model the behavior of markets. In biology, continuity is used to study the growth and development of organisms.