Chapter 8 Continuity
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In mathematics, continuity is a fundamental concept that
describes the behavior of a function. Intuitively, a function is continuous if
it can be drawn without lifting your pen from the paper. More formally, a
function \( f(x) \) is continuous at a point \( x = a \) if three conditions
are satisfied:
1. The function must be defined at \( x = a \), meaning that
\( f(a) \) exists.
2. The limit of the function as \( x \) approaches \( a \)
must exist. Mathematically, this is written as \( \lim_{x \to a} f(x) \).
3. The value of the function at \( x = a \) must be equal to
the limit of the function as \( x \) approaches \( a \). Symbolically, this is
expressed as \( f(a) = \lim_{x \to a} f(x) \).
If all three conditions are met, the function is considered
continuous at \( x = a \).
Additionally, a function is said to be continuous on an
interval if it is continuous at every point within that interval.
Understanding continuity is essential in calculus and analysis as it forms the basis for many important concepts and theorems.