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)
f

^{n}, 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

**d. Exponential function**: if a is positive real number, other than unity, then the function f(x) defined by f(x) = a

^{x}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) = log

_{a}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) = a

_{0}+ a

_{1}x + a

_{2}x + … + a

_{n }x

^{n}, where a

_{0}, a

_{1}, a

_{2},… a

_{n}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.