(1) Suppose f(x) and g(x) be two real functions continuous at a real number a.
Then

(a) f(x) + g(x) is continuous at x=a.

(b) f(x) - g(x) is continuous at x=a.

(c) f(x).g(x) is continuous at x=a.

(d) (f/g)x is continuous at x=a
g(a) ≠ 0

(e) 1/f(x) and 1/g(x) are continuous at n=a, when f(a) + g(a) ≠ 0

(f) If f is a constant function, i.e f(x)=k for some real number k, then the function (k.g)(x) = k.g(x) is also continuous.

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(i) A function is said to be continuous, if it is continuous on the whole of its domain.

(ii) Every identity function is continuous .

(iii) Every constant function is continuous.

(iv) Every polynomial function is continuous.

(v) Every Rational function is continuous.

(vi) All trigonometric functions are continuous in their domain.

(vii) Every Modulus function is continuous.

(viii) For composite function : Suppose f(x) and g(x) are real valued function such that (fog)x is defined at a. If g is continuous at a and if it is continuous at g(a), then (fog)x is continuous at c.