Continuity
A function is continuous at a if f(a) exists, the limit at a exists, and the two are equal. You can draw its graph without lifting your pen.
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Analogy
A road with no potholes or gaps: you can drive across without stopping.
Example
At x = 1 it is undefined (0/0).For x not equal 1: f(x) = x + 1.Limit at 1 is 2.Define f(1) = 2 and it is continuous.
In practice
Continuous functions behave predictably, and many big theorems need continuity to work.
Practice
Flashcards
- What are the three conditions for continuity at a?f(a) is defined, the limit exists, and the limit equals f(a).
Interview questions
- Describe the three types of discontinuity and give an example of each.
- Removable: hole, limit exists, value missing
- Jump: left and right limits differ
- Infinite: function grows without bound
- Examples such as (x^2-1)/(x-1), |x|/x, 1/x^2
Questions people ask
What are the three conditions for continuity at a point?
First, f(a) must be defined. Second, the limit of f(x) as x approaches a must exist. Third, the limit must equal f(a). If any one fails, the function is discontinuous at that point and the graph has a hole, jump or break.
Are polynomials continuous everywhere?
Yes. Polynomials, sine, cosine and exponential functions are continuous at every real number. Rational functions such as 1/x are continuous wherever they are defined, so they break only where the denominator is zero. Sums and products of continuous functions are also continuous.