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# Algebra of Continuous Functions

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#### Algebra of Continuous Functions - Lesson Summary

Let f(x) and g(x) be two real functions continuous at real number c.
lim x â†’ c f(x) = f(c)     lim x â†’ c g(x) = g(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.

lim x â†’ c (f+g)(x) =  lim x â†’ c [f(x) + g(x)]

= lim x â†’ c f(x) +  lim x â†’ c g(x)

= f(c) + g(c) = (f + g)(c)

â‡’ lim x â†’ c (f+g)(x) = (f+g)(c)

Hence, f + g is continuous at x = c.

ii) (f - g)(x) is also defined at c.

lim x â†’ c (fâ€“g)(x) =  lim x â†’ c [f(x) â€“ g(x)]

= lim x â†’ c f(x) â€“  lim x â†’ c g(x) = f(c) - g(c)

= (f - g)(c)

lim x â†’ c (fâ€“g)(x) = (f - g)(c)

Hence, f - g is continuous at x = c.

iii) (f . g)(x) is also defined at c.

lim x â†’ c (f.g)(x) =  lim x â†’ c [f(x) . g(x)]

= lim x â†’ c f(x) .  lim x â†’ c g(x)

= f(c) . g(c) = (f . g)(c)

â‡’ lim x â†’ c (f.g)(x) =  (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)

lim x â†’ c (f/g)(x) =  lim x â†’ c [f(x) / g(x)]

= lim x â†’ c f(x) /  lim x â†’ c g(x)  = 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.

=  lim x â†’ c [f(x) . g(x)] =  lim x â†’ c (k.g)(x) = k .  lim x â†’ c g(x)

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