Definition of Continuity:1) ___________________2) ___________________3) ___________________
Definition of Continuity:
1) A function f(x) is said to be continuous at a point c in its domain if three conditions are satisfied:
– The function is defined at c (i
Definition of Continuity:
1) A function f(x) is said to be continuous at a point c in its domain if three conditions are satisfied:
– The function is defined at c (i.e., f(c) is defined).
– The limit of the function as x approaches c exists.
– The limit of the function as x approaches c is equal to the value of the function at c (i.e., lim (x→c) f(x) = f(c)).
2) A function f(x) is said to be continuous on an interval (a, b) if it is continuous at every point in the interval. This means that the function satisfies the conditions of continuity at each point within the interval.
3) A function f(x) is said to be continuous on a closed interval [a, b] if it is continuous on (a, b) and if the limit of the function exists as x approaches a and x approaches b.
These definitions of continuity provide a formal way of describing the behavior of a function and its smoothness without any abrupt changes or jumps. A continuous function can be graphed without lifting the pen from the paper, as there are no breaks or gaps in the graph. Continuity is an essential concept in calculus and mathematical analysis as it allows for the application of various rules and theorems.
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