# Theorems of Continuity

- Sum, difference, product, and quotient of two continuous functions is always a continuous function. However,
*h*(*x*) =*f*(*x*)/*g*(*x*) is continuous at*x*=*a*only if*g*(*a*) â‰ 0. - If
*f*(*x*) is continuous and*g*(*x*) is discontinuous then*f*(*x*) +*g*(*x*) is a discontinuous function.**Illustration**If*f*(*x*) =*x*and*g*(*x*) = [*x*], greatest integer function.**Sol.**Here*f*(*x*) is continuous at*x*= 0 but*g*(*x*) is discontinuous at*x*= 0.*F*(*x*) =*x*+ [*x*] is discontinuous at*x*= 0 as*f*(0^{+}) = 0 and*f*(0^{â€“}) = â€“1. - If
*f*(*x*) is continuous and*g*(*x*) is discontinuous at*x*=*a*then the product function*h*(*x*) =*f*(*x*) â‹…*g*(*x*) is not necessarily be discontinuous at*x*=*a*.**Illustration**Consider functions*f*(*x*) =*x*^{3}and*g*(*x*) = sgn(*x*).**Sol.**Here*f*(*x*) is continuous at*x*= 0 and*g*(*x*) is discontinuous at*x*= 0.*F*(*x*) =*f*(*x*) â‹…*g*(*x*) = is continuous at*x*= 0. - If
*f*(*x*) and*g*(*x*) are discontinuous at the same point then sum or product of the functions may be continuous. For example,*f*(*x*) = [*x*] (GIF function) and*g*(*x*) = {*x*} (fractional part function) both are discontinuous at*x*= 1 but their sum*f*(*x*) +*g*(*x*) =*x*is continuous at*x*= 1.*f*(*x*) = and*g*(*x*) =*x*= 0, but their product*f*(*x*) Ã—*g*(*x*) = â€“1, âˆ€*x*âˆˆ*R*is continuous at*x*= 0. - Every polynomial is continuous at every point of the real line.
- Every rational function is continuous at every point where its denominator is different from zero.
- Logarithmic functions, exponential functions, trigonometric functions, inverse circular functions, and modulus functions are continuous in their domain.