Continuity is the study of how a function behaves as the input approaches a certain value. A function is continuous at a point if the limit of the function...
Continuity is the study of how a function behaves as the input approaches a certain value. A function is continuous at a point if the limit of the function as it approaches that point is equal to the function value at that point. In other words, the function takes on the same value as its limit at that point.
Continuity has several important properties:
A continuous function is defined at all points in its domain.
A continuous function is differentiable at all points in its domain.
Continuity is equivalent to the limit definition of a derivative.
A function is continuous if and only if its limit exists at that point.
Examples of continuous functions:
All polynomial functions (e.g., f(x) = x^2 + 3)
All rational functions (e.g., f(x) = (x+1)/(x-2))
All exponential functions (e.g., f(x) = e^x)
All trigonometric functions (e.g., f(x) = sin(x))
Examples of functions that are not continuous:
Discontinuities caused by removable singularities
Discontinuities caused by vertical asymptotes
Discontinuities caused by jump discontinuities
Key takeaways:
Continuity is a property of functions that determines their behavior at a given point.
Continuity has several equivalent definitions, including the limit definition, derivative definition, and value definition.
Continuity is a fundamental property in calculus that is used to derive other concepts such as derivatives and integrals