Continuity laws

Summary

• In all laws \(f\) and \(g\) are both continuous at \(x=x_0\)
• Constants are continuous. If \(c\) is a constant then \(f(x)=c\) is continuous on \(R\)
• \(x\) is continuous. \(f(x)=x\) is continuous on \(R\)
• Addition law. \(f+g\) is continuous at \(x=x_0\)
• Subtraction law. \(f-g\) is continuous at \(x=x_0\)
• Constant law. \(cf\) is continuous at \(x=x_0\)
• Multiplication law. \(f⋅g\) is continuous at \(x=x_0\)
• Division law. If \(g(x_0)≠0\) then \(f/g\) is continuous at \(x=x_0\)
• Positive Power law. If \(n\) is a positive integer then \({\left( f\left( x \right) \right)}^n\) is continuous at \(x=x_0\)
• Negative Power law. If \(n\) is a positive integer and \(f(x_0)≠0\) then \({\left( f\left( x \right) \right)}^{-n}\) is continuous at \(x=x_0\)
• Root law. If \(n\) is a positive integer and \(f\left( x_0 \right)≥0\) then \(\sqrt[n]{f(x)}\) is continuous at \(x=x_0\)
• Exponential law. If \(a>0\) then \(a^{f(x)}\) is continuous at \(x=x_0\) ( \(e^x\) is continuous on \(R\) ).
• Composition law). \(g:A→B\) , \(f:B→C\) and \(f∘g:A→C\) is the composition, \((f∘g)(x)=f(g(x))\) . If \(g\) is continuous at \(x=x_0\) and \(f\) is continuous at \(u=g(x_0)\) then \((f∘g)\) is continuous at \(x=x_0\) .
• Inverse law). \(f:A→B\) is bijective and \(f^{-1}:B→A\) is its inverse. If \(f\) is continuous at \(x=x_0\) then \(f^{-1}\) is continuous at \(y=f(x_0)\) ( \(log x\) is continuous on \((0,∞)\) ).