Continuity

A function \(f\) is continuous at \(x=c\) if \(f(c)\) is defined and \(\displaystyle\lim_{x\to c}f(x)=f(c)\). In practice we check three numbers:

\[\text{LHL}=\lim_{x\to c^-}f(x),\qquad \text{RHL}=\lim_{x\to c^+}f(x),\qquad f(c)\]

If all three are equal, \(f\) is continuous at \(c\); otherwise it is discontinuous there. A function is continuous (on an interval) if it is continuous at every point of it.

Building continuous functions

• Polynomials, \(\sin x\), \(\cos x\), \(e^x\) are continuous everywhere; \(\log x\) for \(x>0\); rational functions except where the denominator is 0.
• Sums, differences, products, quotients (denominator \(\neq0\)) and compositions of continuous functions are continuous.
• \(|x|\) is continuous everywhere; the greatest integer function \(\lfloor x\rfloor\) is discontinuous at every integer.

Piecewise functions. Each piece is usually continuous on its own interval, so only the "joining points" need checking. To make \(f(x)=\begin{cases}kx+1,&x\le2\\3x-1,&x>2\end{cases}\) continuous, equate the one-sided limits at 2: \(2k+1=5\Rightarrow k=2\).