Continuous Function
a function that can be graphed with a line or a smooth curve
A continuous function is a type of function in mathematics that retains a constant graph, without any jumps or gaps, over the entire domain. In other words, a function is continuous if it can be drawn as a smooth curve without lifting up the pencil.
Formally, a function f(x) is said to be continuous at point c if the limit of f(x) as x approaches c is equal to f(c), i.e.,
lim x→c { f(x) } = f(c)
A function is said to be continuous on an interval [a, b] if it is continuous at every point in the interval.
Intuitively, a continuous function is one that can be drawn without lifting the pencil at any point, and can be visualized as a smooth curve without breaks, holes, or jumps. For example, the function f(x) = x² is continuous because it can be drawn as a smooth curve without any gaps or breaks, whereas the function f(x) = 1/x is not continuous at x = 0 because it has a vertical asymptote there.
Continuous functions are essential in most areas of mathematics, including calculus, analysis, and topology. They have numerous applications in physics, engineering, economics, and other fields.
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