To turn in (due Monday, together with 2.3): 2.4 4, 12, 52
Notes
Continuity
Continuity is meant to describe the concept that we can draw the graph of a function without lifting our pen: The function forms a continuous whole.
The idea is to relate the limit of a function at a point \(a\) to the value of the function at \(a\). The limit tells us what happens near the point \(a\). For continuity, this must agree with what happens exactly at \(a\).
Definition:
We say that a function \(f(x)\) is continuous at a point \(x=c\) if:
The limit \(\displaystyle\lim_{x\to a}f(x)\) exists.
The value \(f(a)\) exists.
They agree: \(\displaystyle\lim_{x\to a}f(x) = f(a)\).
Practical implication: Can compute a limit by “plugging in”.
If this is not the case, we say that the function is discontinuous at \(x=c\).
We say that \(f\) is continuous on an interval\([a,b]\), if it is continuous at every point of the interval.
Examples of continuous functions:
Constant function
\(x^n\)
Sum of continuous functions is continuous
Product of continuous functions is continuous
Quotient of continuous function is continuous, except where the denominator is \(0\)
Trigonometric functions are continuous
Possible discontinuities:
Left-sided limit and right-sided limit exist but differ (jump discontinuity)
Limit exists, but value is different from it (removable discontinuity)
One-sided limits are infinity (infinite discontinuity) or don’t exist
Example of studying the continuity of a piecewise-defined function (example 2)
Function composition: If \(g(x)\) is continuous at \(x=c\), and \(f(u)\) is continuous at \(u=g(c)\), then \(f(g(x))\) is also continuous at \(x=c\).