Continuity
Continuity Conditions (At a Point)
Continuity Conditions (At a Point)
For a function to be continuous at a point (x = a), 3 conditions must be met:
For a function to be continuous at a point (x = a), 3 conditions must be met:
(i) The left-hand limit and the right-hand limit of the function must tend to the same limit, L.
(i) The left-hand limit and the right-hand limit of the function must tend to the same limit, L.
(ii) The function must be defined at x = a.
(ii) The function must be defined at x = a.
(iii) f(a) = L.
(iii) f(a) = L.
Discontinuities
Discontinuities
There are 3 + 1 types of discontinuities for a function.
There are 3 + 1 types of discontinuities for a function.
(i) Asymptotic Discontinuity (Infinite Discontinuity): This occurs when either the Left-hand limit or the right-hand limit of the function at that point tends to positive or negative infinity.
(i) Asymptotic Discontinuity (Infinite Discontinuity): This occurs when either the Left-hand limit or the right-hand limit of the function at that point tends to positive or negative infinity.
(ii) Jump Discontinuity: This occurs when the left-hand limit and right hand limit are real numbers but they are NOT equal.
(ii) Jump Discontinuity: This occurs when the left-hand limit and right hand limit are real numbers but they are NOT equal.
(iii) Removable Discontinuity: This occurs when the left-hand limit and right hand limit are equal for a given x-value but the function is not defined or the function's output is not equal to the limit, and
(iii) Removable Discontinuity: This occurs when the left-hand limit and right hand limit are equal for a given x-value but the function is not defined or the function's output is not equal to the limit, and
(iv) Endpoints for functions defined on a bounded interval
(iv) Endpoints for functions defined on a bounded interval
Move the magnifying glass in the applet below to learn more about them.
Move the magnifying glass in the applet below to learn more about them.
The Epsilon-Delta Definition
The Epsilon-Delta Definition
The function, f, is continuous at c if for every ε > 0, there exist δ > 0 such that if |x - c| < δ then |f(x) - f(c)| < ε.
The function, f, is continuous at c if for every ε > 0, there exist δ > 0 such that if |x - c| < δ then |f(x) - f(c)| < ε.