In the previous section, we saw how to check whether a given function is a continuous function or not. In this section, we will see algebra of continuous functions.
• In class 11, we have seen algebra of limits. (Details here). Now in the previous few sections, we saw that, limits are the deciding factors for continuous functions. So naturally, we can think about algebra of continuous functions.
• Suppose that, f and g are two real functions.
• Also suppose that:
♦ f is continuous at c.
♦ g is continuous at c.
• Then we can write four results:
♦ f+g is continuous at x = c
♦ f−g is continuous at x = c
♦ f.g is continuous at x = c
♦ $\frac{f}{g}$ is continuous at x = c (provided g(c) ≠ 0)
• We will write the proof for the first result. It can be written in 4 steps:
1. We want to prove that (f+g) is continuous at c.
2. Let us apply the first condition.
That is: $\lim_{x\rightarrow c} (f+g)(x)$ must exist.
• Let us check whether this is true.
$\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{\lim_{x\rightarrow c} (f+g)(x)} & {~=~} &{\lim_{x\rightarrow c} [f(x) + g(x)]} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{\lim_{x\rightarrow c} f(x)~+~\lim_{x\rightarrow c} g(x)} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{f(c) ~+~ g(c)} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{(f+g)(c)} \\
\end{array}$
◼ Remarks:
• 1 (magenta color). Here we use the fact about addition of functions.
(f+g)(x) = f(x) + g(x)
2 (magenta color). Here we use the algebra of limits that we saw in class 11.
3 (magenta color). We get this result because, it is given that, f and g are continuous at c.
4 (magenta color). Here we again use the fact about addition of functions.
(f+g)(x) = f(x) + g(x)
So $\lim_{x\rightarrow c} (f+g)(x)$ exists.
3. Let us apply the second condition:
$\lim_{x\rightarrow c} (f+g)(x)$ must be equal to (f+g)(c)
This is already proved in (2) above.
4. Since both conditions are satisfied, (f+g)(x) is continuous at c.
The proofs for the remaining three results can be written in a similar way. The reader is advised to write those proofs in his/her own notebooks.
Now we will see two special cases.
Case 1
This can be written in 4 steps.
1. Consider result 3:
If both f and g are continuous at c, then f.g is also continuous at c.
2. Suppose that, one of the two functions, say f, is a constant function. Then we can write: f(x) = λ, where λ is a real number.
(Recall that, any constant function is a continuous function)
3. Then we can write (λ.g)(x) is a continuous function.
• This is same as: λ[g(x)] is a continuous function.
(We saw this result in class 11, in the topic of multiplication of functions)
4. From this, we get an interesting result:
If λ = −1, then −g(x) is a continuous function.
• So, if g is a continuous function, then -g is also a continuous function.
Case 2
This can be written in 4 steps.
1. Consider result 4:
If both f and g are continuous at c, then $\frac{f}{g}$ is also continuous at c.
2. Suppose that, f is a constant function. Then we can write: f(x) = λ, where λ is a real number.
(Recall that, any constant function is a continuous function)
3. Then we can write $\left( \frac{\lambda}{g} \right) (x)$ is a continuous function.
• This is same as: $\lambda \left[\left( \frac{1}{g} \right) (x) \right]$ is a continuous function.
4. From this, we get an interesting result:
If λ = 1, then $\left( \frac{1}{g} \right) (x)$ is a continuous function.
• So, if g is a continuous function, then $\frac{1}{g}$ is also a continuous function.
This case can be used to solve many problems.
We have seen the algebra of limits. Using those results, we can prove that, any polynomial function is continuous. It can be written in steps:
1. We want to prove that
$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ~.~.~.~+ a_n x^n$
is continuous.
• We can use mathematical induction.
2. First, we check whether the function is continuous when n = 1.
• That is., we want to know whether $f(x) = a_0 + a_1 x$ is continuous.
• It is indeed continuous because, it is the sum of two continuous functions: $f_1 (x) = a_0 $ and $f_2 (x) = a_1 x$
3. Next, we assume that, the function is continuous when n = k.
This can be written as follows:
$\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{f(x)} & {~=~} &{a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ~.~.~.~+ a_k x^k} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{a_0 + x(a_1 + a_2 x^1 + a_3 x^2 + ~.~.~.~+ a_{k} x^{k-1})} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{f_1 (x) + x[f_3 (x)]} \\
\end{array}$
• So f(x) is the sum of two functions:
(i) $f_1 (x) = a_0 $
(ii) $x[f_3 (x)] = x(a_1 + a_2 x^1 + a_3 x^2 + ~.~.~.~+ a_{k} x^{k-1}) $
• We assume that, this f(x) is continuous. If f(x) is continuous, then it's components will also be continuous. That is.,
♦ $f_1 (x)$ is continuous.
♦ $x[f_3 (x)]$ is continuous.
• If $x[f_3 (x)]$ is continuous, then $f_3 (x)$ will be continuous.
• So, by assuming that the function is continuous when n = k, we get an important result:
$f_3 (x)$ is continuous.
4. Next, we check whether the function is continuous when n = k+1.
This can be written as follows:
$\begin{array}{ll} {~\color{magenta} 1 } &{{}} &{f(x)} & {~=~} &{a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ~.~.~.~+ a_k x^k + a_{k+1} x^{k+1}} \\
{~\color{magenta} 2 } &{{}} &{{}} & {~=~} &{a_0 + x(a_1 + a_2 x^1 + a_3 x^2 + ~.~.~.~+ a_{k} x^{k-1} + a_{k+1} x^k)} \\
{~\color{magenta} 3 } &{{}} &{{}} & {~=~} &{a_0 + x(a_1 + a_2 x^1 + a_3 x^2 + ~.~.~.~+ a_{k} x^{k-1}) + x \, a_{k+1} x^k} \\
{~\color{magenta} 4 } &{{}} &{{}} & {~=~} &{f_1 (x) + x[f_3 (x)] + a_{k+1} x^{k+1}} \\
\end{array}$
• So this f(x) is the sum of three components.
(i) We already know that, $f_1 (x)$ is continuous.
(ii) We already know that, $x[f_3 (x)]$ is continuous.
(iii) The third component is also continuous because, it is the product of $a_{k+1}$ and x taken (k+1) times.
• So f(x) is continuous.
♦ That means, the function is continuous when k = 1.
♦ Also, if the function is continuous for n = k, it will be continuous for n = (k+1)
♦ So we prove the continuity by mathematical induction.
In the next section, we will see some solved examples.
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