Understanding Continuity | A Fundamental Concept in Calculus and Real Analysis

definition of continuity: f is continuous at c iff

the limit of f(x) as x approaches c exists and is equal to f(c)

the limit of f(x) as x approaches c exists and is equal to f(c). In other words, a function f is continuous at a point c if the function is defined at that point, the limit of the function as x approaches c exists, and the value of the function at c is equal to the limit.

This definition can be written mathematically as:
f is continuous at c if lim(x->c) f(x) = f(c).

Continuity is an important concept in calculus and real analysis. It ensures that there are no abrupt jumps, holes, or breaks in the graph of a function. When a function is continuous at each point in its domain, it can be smoothly drawn without lifting the pencil from the paper.

There are three types of continuity:

1. Pointwise Continuity: A function f is said to be pointwise continuous if it is continuous at each individual point in its domain.

2. Uniform Continuity: A function f is uniformly continuous if the rate of change of the function can be controlled uniformly over its entire domain. In other words, for any given epsilon > 0, there exists a delta > 0 such that |f(x) – f(y)| < epsilon whenever |x - y| < delta. 3. Global Continuity: A function f is said to be globally continuous if it is both pointwise continuous and uniformly continuous over its entire domain. Continuity plays a crucial role in many areas of mathematics and its applications. It allows us to make precise calculations and analyze the behavior of functions in a systematic way.

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