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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 โ h) = \(\displaystyle{\lim_{h \to 0}}\) f(a + h) = f(a), h > 0
i.e. \(LHL_{x = a}\) = \(RHL_{x = a}\) equals value of โfโ 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
ย ย \(\displaystyle{\lim_{x \to {1^-}}}\) 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 ย f(1) = \(\displaystyle{\lim_{x \to {1^-}}}\) f(x) = \(\displaystyle{\lim_{x \to {1^+}}}\) f(x)
\(\therefore\) ย 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 โaโ i.e. \(\displaystyle{\lim_{x \to {a^+}}}\) f(x) = f(a) = a finite quantity.
(iii) f is left continuous at โbโ 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!
