A function \(f(x)\) is said to be continuous at the point \(x=c\) contained within the interval \((a,b)\) provided that the three conditions hold. 

1. The limit exists at \(x=c\), i.e. \(\underset{x\to c^-}{\lim}f(x)=\underset{x\to c^+}{\lim}f(x)=L \)
2. The function \(f(x)\) exists at \(x=c\), i.e. \(f(c)\) exists.
3. The value \(f(c)\) equals to \(L\)

If \(f(x)\) is continuous at \(x=c\), then \(\underset{x\to c}{\lim}f(x)=f(c)\)

Note: There are some functions such as polynomials, exponentials, cosine, and sine which are continuous everywhere.

Intermediate Value Theorem

Let \(f(x)\) be a continuous function on the closed interval \([a,b]\) and let \(N\) be a number such that \(f(a)<N<f(b)\) and \(f(a)\neq f(b)\) then there exists a number \(c\) such that \(a<c<b\) in which \(f(c)=N\)

Intermediate Value Theorem (Corollary)

Let \(f(x)\) be a continuous function on the closed interval \([a,b]\) such that \(f(a)<0<f(b)\) or \(f(b)<0<f(a)\) then there exists a number \(c\) such that \(f(c)=0\)