# Definition of Continuity of a Function

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## Continuous Functions

A Function for which a small change in the independent variable causes only a small change and not a sudden jump in the dependent variable are called continuous functions. Naively, we may say that a function is continuous at a fixed point if we can draw the graph of the function around that point without lifting the pen from the plane of the paper.

## Continuity of a function at a point

A function f(x) is said to be continuous at x = a, if

$$\displaystyle{\lim_{x \to a}}$$ f(x) = f(a).

Symbolically f is continuous at x = a if

$$\displaystyle{\lim_{h \to 0}}$$ f(a – h) = $$\displaystyle{\lim_{h \to 0}}$$ f(a + h) = f(a), h > 0

i.e. $$LHL_{x = a}$$ = $$RHL_{x = a}$$ equals value of ‘f’ at x = a.

Example : If f(x) = {$$sin{\pi x\over 2}$$, x < 1 and [x], x $$\geq$$ 1} then find whether f(x) is continuous or not at x = 1, where [ ] denotes greatest integer function.

Solution : For continuity at x = 1, we determine, f(1), $$\displaystyle{\lim_{x \to {1^-}}}$$ f(x) and $$\displaystyle{\lim_{x \to {1^+}}}$$ f(x)

Now, f(1) = [1] = 1

$$\displaystyle{\lim_{x \to {1^-}}}$$ f(x) = $$\displaystyle{\lim_{x \to {1^-}}}$$ $$sin{\pi x\over 2}$$ = $$sin{\pi\over 2}$$ = 1 and $$\displaystyle{\lim_{x \to {1^+}}}$$ f(x) = $$\displaystyle{\lim_{x \to {1^+}}}$$ [x] = 1

so   f(1) = $$\displaystyle{\lim_{x \to {1^-}}}$$ f(x) = $$\displaystyle{\lim_{x \to {1^+}}}$$ f(x)

$$\therefore$$   f(x) is continuous at x = 1.

## Continuity of a function in an interval

(a) A function is said to be continuous in (a,b) if f is continuous at each & every point belonging to (a, b).

(b) A function is said to be continuous in a closed interval [a,b] if :

(i) f is continuous in the open interval (a,b)

(ii) f is right continuous at ‘a’ i.e. $$\displaystyle{\lim_{x \to {a^+}}}$$ f(x) = f(a) = a finite quantity.

(iii) f is left continuous at ‘b’ i.e. $$\displaystyle{\lim_{x \to {b^-}}}$$ f(x) = f(b) = a finite quantity.

Note :

(i) All polynomials, trigonometrical functions, exponential & logarithmic functions are continuous in their domains.

(ii) If f(x) & g(x) are two functions that are continuous at x = c then the function defined by:

$$F_1(x)$$ = f(x) + g(x); $$F_2(x)$$ = Kf(x), where K is any real number; $$F_3(x)$$ = f(x).g(x) are also continuous at x = c.

Further, if g(c) is not zero, then $$F_4(x)$$ = $$f(x)\over g(x)$$ is also continuous at x = c.

Hope, you learnt definition of continuity of a function and continuity of a function at a point and over an interval. Practice more question on continuity of a function to learn more and get ahead in competition. Good Luck!