Askiitians Tutor Team
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In mathematics, a continuous function is a function that does not have any abrupt changes or breaks in its graph. It is a fundamental concept in calculus and analysis.
Formally, a function f(x) is said to be continuous at a point x = a if three conditions are satisfied:
The function is defined at a: f(a) is a well-defined real number.
The limit of the function as x approaches a exists: lim┬(x→a)〖f(x)〗 exists and is a real number.
The limit of the function as x approaches a is equal to the value of the function at a: lim┬(x→a)〖f(x)〗=f(a).
In simpler terms, a function is continuous if you can draw its graph without lifting your pen from the paper. This means that there are no holes, jumps, or vertical asymptotes in the graph. Intuitively, the function changes smoothly and gradually as the input values change.
Continuous functions have many important properties. For example, if a function is continuous on a closed interval, then it is guaranteed to have a maximum and minimum value on that interval. In calculus, the concept of continuity is essential for defining important notions such as derivatives and integrals.
It's worth noting that not all functions are continuous. Discontinuous functions may have points of discontinuity, such as jump discontinuities, removable discontinuities, or essential discontinuities, where the conditions for continuity mentioned earlier are not met.