A function f(x) is said to be continuous, if it is continuous at each point of its domain.
Everywhere continuous function: A function f(x) is said to be everywhere continuous if it is continuous on the entire real line (-∞, ∞) i.e. On R.
Some fundamental results on continuous functions: Here, we list some fundamentals result on continuous functions without giving their proofs.
Result: Let f (x) and g(x) be two continuous functions on their common domain D and let c be real number. Then
(i) C f is continuous
(ii) f + g is continuous
(iii) f - g is continuous
(iv) fg is continuous
(v) f/g is continuous
(vi) fn, for all n ϵ N is continuous.
Result: Listed below are some common type of functions that are continuous in their domains.
a. Constant function: Every constant function is every – where continuous.
b. Identity function: The identity function I(x) is defined by I(x) = x for all x ϵ R
c. Modulus function: The modulus function f (x) is defined as
clearly, the domain of f(x) is R and this function is everywhere continuous.
d. Exponential function: if a is positive real number, other than unity, then the function f(x) defined by f(x) = ax for all x ϵ R is called the exponential function. The domain of this function id R. it is evident form its graph that it everywhere continuous.
e. Logarithm function: if a is positive real number other than unity, then a function by f(x) = loga x is called the logarithm function. Clearly its domain is the set of all positive real numbers and it is continuous on its domain.
f. Polynomial function: A function of the form f (x) = a0 + a1x + a2x + … + an xn, where a0, a1, a2,… an ϵ R is called a polynomial function. This function is everywhere continuous.
g. Rational function: if p(x) and q(x) are two polynomials, then a function, f(x) of the form f(x) = p(x)/q(x), q(x) ≠ 0 is called a polynomial function. This function is continuous on its domain i.e., it is everywhere continuous except at points where q(x) = 0.
h. Trigonometric functions: all trigonometrical functions viz. sin x, cos x, tan x, cosec x, cot x are continuous at each point of their respective domains.
Result: The composition of two continuous functions is a continuous function i.e., f and g are two functions such that g is continuous at a point a and f is continuous at g (a), then fog continuous at a.