Continuity at a Point and on an Interval
f is continuous at a when f(a) exists, the limit at a exists, and the two are equal.
What is Continuity at a Point and on an Interval?
f is continuous at a when f(a) exists, the limit at a exists, and the two are equal.
Key formula / rule: All three conditions must hold; failure of any one gives a discontinuity.
Key points
- Evaluate continuity of a piecewise function at its junction points.
- Classify the type of discontinuity at a given point.
Common exam trap
Checking only that the two one-sided limits agree and ignoring f(a).
Definitions
- Term
Continuity at a Point and on an Interval
- Meaning
f is continuous at a when f(a) exists, the limit at a exists, and the two are equal.
- Term
Continuity at a Point and on an Interval — explanation
- Meaning
Continuity is a three-part condition, and questions are usually built by breaking exactly one part. Piecewise definitions and parameter matching at junction points are the standard examination format.
Learning objectives
Evaluate continuity of a piecewise function at its junction points.
Classify the type of discontinuity at a given point.
Formulae
- Key point
All three conditions must hold; failure of any one gives a discontinuity.
- Key point
Removable, jump and infinite discontinuities are distinguished by which condition fails.
- Key point
Sums, products and compositions of continuous functions are continuous.
Prerequisites
MAT-U2-LC-T1-S1-C1
Common mistakes
Checking only that the two one-sided limits agree and ignoring f(a).
Assuming a function defined by a single formula is continuous everywhere in its domain of writing.
Keywords
Continuity
Point
Interval
Practice preview
Consider the function f(x) defined as: f(x) = { x^2 + 1, if x <= 0 ; x + 1, if x > 0 }. Is f(x) continuous at x = 0?…
easy
For which of the following functions is the function f(x) = sin(x) + x^3 continuous?…
easy
If the function f(x) = { ax + 1, if x <= 3 ; bx + 3, if x > 3 } is continuous at x = 3, what is the relation between a and b?…
medium