Suppose that \(f\) is a function and \(a\) is a number. We say that \(f\) is continous at \(a\) if \(f(a)\) is defined, \(\lim_{x \to a} f(x)\) is defined and \(\lim_{x \to a} f(x) = f(a)\).

We say that a function \(f\) is continous on open interval \(I\) if for every \(a \in I\), \(f\) is continous at a.

Suppose \(f\) and \(g\) are continous at \(a\). Then so are \(f + g\), \(f - g\), and \(f.g\). If \(g(a) \neq 0\), then \(f/g\) is also continous at a.

We say that \(f\) is continous from the right at \(a\) if \(f(a)\) is defined, \(\lim_{x \to a^+} f(x)\) is defined, and \(\lim_{x \to a^+} f(x) = f(a)\). We say that \(f\) is continous from the left at \(a\) if \(f(a)\) is defined, \(\lim_{x \to a^-} f(x)\) is defined, and \(\lim_{x \to a^-} f(x) = f(a)\).

We say that a function \(f\) is continous on a closed interval \([a,b]\) if it is continous on \((a,b)\), continous fom the right at \(a\), and continous from the left at \(b\).

Suppose that \(f\) is continous on the interval \([a,b]\) and either \(f(a) < r < f(b)\) or \(f(a) > r > f(b)\). Then there is some number \(c\) such that \(a < c < b\) and \(f(c) = r\).

Let \(n\) be any positive integer and let \(f(x) = \sqrt[n]{x}\)

• If \(n\) is even, then \(f\) is continous on \([0, \infty)\)
• If \(n\) is odd, then \(f\) is continous on \((-\infty, \infty)\)

For any numbers \(\alpha\) and \(\beta\),

\(|cos \beta - cos \alpha| \leq | \beta - \alpha|\), \(|sin \beta - sin \alpha| \leq | \beta - \alpha|\)

All of the trignometric functions are continous everywhere they are defined.

Suppose that \(f\) is continous at \(a\). Then as \(x \to a\), \(f(x) \to f(a)\). In other words, for every \(\epsilon > 0\) there is some \(\delta > 0\) such that if \(|x - a| < \delta\) then \(|f(x) - f(a)| < \epsilon\)

• If \(f\) is continous from the left at \(a\), then as \(x \to a^{\leq}\), \(f(x) \to f(a)\)
• If \(f\) is continous from the right at \(a\), then as \(x \to a^{\geq}\), \(f(x) \to f(a)\)

1. Suppose that \(\lim_{x \to a} g(x) = L\) and \(f\) is continous at \(L\). Then \(\lim_{x \to a} f(g(x)) = f(L)\).
2. Supppose that \(g\) is continous at \(a\) and \(f\) is continous at \(g(a)\). Then \(f \circ g\) is continous at \(a\).
3. Suppose that \(g\) is continous on an interval \(J\), \(f\) is continous on an interval \(I\), and for every \(x \in J\), \(g(x) \in I\). Then \(f \circ g\) is continous on \(J\).