Notes On Algebra of Continuous Functions - CBSE Class 12 Maths
Let f(x) and g(x) be two real functions continuous at real number c. $\underset{xâ†’\text{c}}{\text{lim}}\text{f(x) = f(c)}$   . Now check the continuity of the sum, difference, product and quotient of these functions f + g, f - g, f . g and f/g Since f(x) and g(x) are continuous at c, f(x) and g(x) are defined at c. i) (f + g)(x) is also defined at c. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) + g(x)]}$$\text{}$ = $\text{}$ = f(c) + g(c) = (f + g)(c) â‡’ Hence, f + g is continuous at x = c. ii) (f - g)(x) is also defined at c. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) â€“ g(x)]}$$\text{}$ = $\text{}$ = f(c) - g(c) = (f - g)(c) $\underset{xâ†’\text{c}}{\text{lim}}\text{(fâ€“g)(x) =}$ (f - g)(c) Hence, f - g is continuous at x = c. iii) (f . g)(x) is also defined at c. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) . g(x)]}$$\text{}$ = = f(c) . g(c) = (f . g)(c) â‡’ (f.g)(c) Hence, f . g is continuous at x = c. iv) f(x)/g(x), g(c) â‰  0 is also defined at c. (f/g)(x) = f(x)/g(x) $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) / g(x)]}$$\text{}$ = = f(c)/g(c) = (f/g)(c) Hence, f/g and g â‰  0 is continuous at x = c. Special case: f(x) = k, where k is a real number. â‡’ f(x). g(x) = (k . g)(x) = k . g(x) is also continuous. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) . g(x)]}$$\text{}$ $\text{}$ = k .$\underset{xâ†’\text{c}}{\text{lim}}\text{g(x)}$$\text{}$ If f(x) is continuous at c and k âˆˆ R, then k. f(x) is also continuous at c. k = -1 â‡’ -f is continuous. If f(x) is continuous at c the, then -f(x) is also continuous at c. â‡’ f(x)/g(x) = k/g(x) is also continuous. If g(x) is continuous at c and k âˆˆ R, then k/g(x) is also continuous at c provided g(c) â‰  0. k = 1 â‡’ 1/g(x) is continuous. The reciprocal of a continuous function is also continuous

#### Summary

Let f(x) and g(x) be two real functions continuous at real number c. $\underset{xâ†’\text{c}}{\text{lim}}\text{f(x) = f(c)}$   . Now check the continuity of the sum, difference, product and quotient of these functions f + g, f - g, f . g and f/g Since f(x) and g(x) are continuous at c, f(x) and g(x) are defined at c. i) (f + g)(x) is also defined at c. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) + g(x)]}$$\text{}$ = $\text{}$ = f(c) + g(c) = (f + g)(c) â‡’ Hence, f + g is continuous at x = c. ii) (f - g)(x) is also defined at c. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) â€“ g(x)]}$$\text{}$ = $\text{}$ = f(c) - g(c) = (f - g)(c) $\underset{xâ†’\text{c}}{\text{lim}}\text{(fâ€“g)(x) =}$ (f - g)(c) Hence, f - g is continuous at x = c. iii) (f . g)(x) is also defined at c. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) . g(x)]}$$\text{}$ = = f(c) . g(c) = (f . g)(c) â‡’ (f.g)(c) Hence, f . g is continuous at x = c. iv) f(x)/g(x), g(c) â‰  0 is also defined at c. (f/g)(x) = f(x)/g(x) $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) / g(x)]}$$\text{}$ = = f(c)/g(c) = (f/g)(c) Hence, f/g and g â‰  0 is continuous at x = c. Special case: f(x) = k, where k is a real number. â‡’ f(x). g(x) = (k . g)(x) = k . g(x) is also continuous. $\underset{xâ†’\text{c}}{\text{lim}}\text{[f(x) . g(x)]}$$\text{}$ $\text{}$ = k .$\underset{xâ†’\text{c}}{\text{lim}}\text{g(x)}$$\text{}$ If f(x) is continuous at c and k âˆˆ R, then k. f(x) is also continuous at c. k = -1 â‡’ -f is continuous. If f(x) is continuous at c the, then -f(x) is also continuous at c. â‡’ f(x)/g(x) = k/g(x) is also continuous. If g(x) is continuous at c and k âˆˆ R, then k/g(x) is also continuous at c provided g(c) â‰  0. k = 1 â‡’ 1/g(x) is continuous. The reciprocal of a continuous function is also continuous

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