Rough introduction to continuity

Summary

Suppose that \(f\) is defined for all \(x\) "near" \(x_0\) (including \(x_0\) ).
\(f\) is continuous at \(x=x_0\) if a small change in \(x\) around \(x=x_0\) results in a small change in \(y\) .
\(f\) is continuous if it is continuous at every point in its domain.
If the domain of \(f\) is a closed interval then \(f\) is continuous if its graph is connected (no breaks or jumps).
Examples:

\(f(x)=2x+1\) is continuous at \(x=0\) . Further, it is continuous at every point in \(R\) so it is continuous.
\(f\) defined as \(f\left( x \right)=0\) if \(x<0\) and \(f\left( x \right)=1\) if \(x≥0\) is not continuous at \(x_0=0\) . \(f(0)=1\) while \(f(-0.001)=0\) ; a small change in \(x\) leads to a large change in \(y\) . The function is continuous at the remaining points in the domain.
If \(f(x)= \frac{x^2-1}{x-1}\) then \(f\) is not continuous at \(x_0=1\) since \(x_0\) is not in the domain of \(f\) .

\(f:A→B\) is continuous at \(x=x_0\) if it is defined at \(x=x_0\) and \({lim}_{x→x_0}f(x)=f(x_0)\)

\(f:A→B\) is continuous at \(x=x_0∈A\) if \(x_0\) is a limit point of \(A\) and for each \(ε>0\) , there exists a \(δ>0\) such that \(|f(x)f(x_0)|<ε\) for all \(x∈A\) such that \(|x-x_0|<δ\) .