CHANGE AND LIMITS 09
Continuity
A function is continuous at a point when its nearby values approach its actual value there.
The main idea
A function \(f\) is continuous at \(x=a\) when
So the limit and the actual function value agree.
1. Three conditions for continuity
For \(f\) to be continuous at \(x=a\), three things must hold.
First, the function must be defined at \(a\):
\[ f(a)\text{ exists.} \]Second, the limit must exist:
\[ \lim_{x\to a}f(x)\text{ exists.} \]Third, they must be equal:
\[ \lim_{x\to a}f(x)=f(a). \]2. A simple continuous function
Consider
\[ f(x)=x^2. \]At \(x=2\),
\[ f(2)=4. \]Also,
\[ \lim_{x\to2}x^2=4. \]Therefore,
\[ \lim_{x\to2}f(x)=f(2), \]so \(f\) is continuous at \(x=2\).
3. Continuous on an interval
If a function is continuous at every point of an interval, we say it is continuous on that interval.
Polynomial functions are continuous for all real \(x\).
For example,
\[ f(x)=x^3-2x+1 \]is continuous on
\[ (-\infty,\infty). \]4. A removable discontinuity
Consider
\[ f(x)=\frac{x^2-1}{x-1}. \]For \(x\neq1\),
\[ f(x)=x+1. \]So
\[ \lim_{x\to1}f(x)=2. \]But the original function is undefined at \(x=1\).
Therefore the function is not continuous there.
5. Filling the hole
If we define
\[ f(1)=2, \]then
\[ \lim_{x\to1}f(x)=f(1)=2. \]The discontinuity has been removed.
This is why it is called a removable discontinuity.
6. A jump discontinuity
Suppose
\[ f(x)= \begin{cases} 1, & x<0,\\ 3, & x\ge0. \end{cases} \]Then
\[ \lim_{x\to0^-}f(x)=1 \]and
\[ \lim_{x\to0^+}f(x)=3. \]Since the two one-sided limits are different,
\[ \lim_{x\to0}f(x) \]does not exist.
So the function is not continuous at \(x=0\).
7. Infinite discontinuity
Consider
\[ f(x)=\frac{1}{x}. \]At \(x=0\), the function is undefined.
As \(x\) approaches \(0\), the function becomes unbounded.
\[ \frac1x\to\infty \quad\text{or}\quad -\infty \]depending on the direction of approach.
This gives an infinite discontinuity.
8. Visual idea
9. Continuous does not mean straight
A continuous graph may be curved.
For example,
\[ y=x^2 \]is continuous even though its slope changes.
Continuity concerns breaks or gaps, not whether the graph is straight.
10. Continuous does not mean differentiable
A function can be continuous at a point but still fail to have a derivative there.
For example,
\[ f(x)=|x| \]is continuous at \(x=0\), but its graph has a sharp corner there.
The left and right slopes are different, so the derivative at \(0\) does not exist.
11. Why continuity matters for calculus
Differentiability requires continuity.
But the reverse is not always true.
Continuity therefore provides an important bridge between limits and derivatives.
12. The next step
We now understand finite change, average rate of change, secant lines, limits and continuity.
The next idea is to let the secant line approach a tangent line and define the derivative from first principles.