A function \(f\) is continuous at a number \(a\) if
\[
\lim_{x\to a}f(x)=f(a)
\]
Otherwise, we say \(f\) is discontinuous at \(a\)
Removable discontinuity (Hole) | Jump discontinuity (Jump) | Essential discontinuity |
---|---|---|
Essential discontinuity - when one of onsided limits does not exist or infinite.
Where are each of the following functions discontinuous?
\( f(x)=\frac{x^2-x-2}{x-2}\)
\( f(x)= \left\{ \begin{array}{ll} \frac{1}{x} & x\neq 0\\ 1 & x=0 \end{array} \right. \)
3. \( f(x)= \left\{ \begin{array}{ll} \frac{x^2-x-2}{x-2}& x\neq 2\\ 1 & x=2 \end{array} \right. \)
A function \(f\) is continuous from the right at a number \(a\) if
\[ \lim_{x\to a^+} f(x)=f(a)\]
A function \(f\) is continuous from the left at a number \(a\) if
\[ \lim_{x\to a^-} f(x)=f(a)\]
A function \(f\) is continous on an interval if it is continuous at every number in the interval(interior).
If \(f\) is defined only on one side of an endpoint of the interval(boundary),
we understand continous at the endpoint to mean continuous from the right or continuous from the left.
We should handle two endpoints carefully.
Check whether the function \(f(x)=1-\sqrt{1-x}\) is continuous on the interval \([-1,1]\).
(Interior points) \(-1 < a < 1\)
(End points) \( a=-1 \) or \( a=1 \)
If \(f\) and \(g\) are continuous at \(a\) and \(c\) is a constant, then the following functions are also continuous at \(a\),
\[ \lim_{x\to -2} \frac{x^3+2x^2-1}{5-3x} \]
On what intervals is each function continuous?
\[ \lim_{x\to \pi} \frac{\sin{x}}{2\cos{x}}\]
If \(f\) is continuous at \(b\) and \(\lim_{x\to a} g(x)=b\), then \[ \lim_{x\to a}f(g(x)) = f(\lim_{x\to a} g(x))=f(b)\] In other words, a evaluation of a continuous function commutes with the limit process.
\[ \lim_{x\to a} \sqrt[n]{g(x)}=\sqrt[n]{\lim_{x\to a}g(x)} \]
if \(\lim_{x\to a} g(x)=b~>0\)
If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a)\), then the composite function \(f\circ g\) is continuous at \(a\).
Where are the following functions continuous?