logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
0 votes

True or False: The composition of two continuous functions is a continuous function.

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • A function is said to be continuous on an open interval $(a,b)$ if it is continuous at every point on the interval $(a,b)$.$\mid f\mid$ is defined as $\mid f\mid(x)=\mid f(x)\mid$
Step 1:
Let $f$ and $g$ be two real functions such that $gof$ exists.
Then Range(f) $\leq$ Domain(g)
Let $'a'$ be an arbitrary point in the domain of $f$.
Then $a\in$ Domain (f) and $f(a)\in $ Domain (g).
$\Rightarrow f$ is continuous at $x=a$ and $g$ is continuous at $f(a)$.
$\Rightarrow \lim\limits_{\large x\to a}f(x)=f(a)$ and $\lim\limits_{\large y\to a}g(y)=g(f(a))$
$\Rightarrow \lim\limits_{\large x\to a}f(x)=f(a)$ and $\lim\limits_{\large f(x)\to f(a)}g(f(x))=g(f(a))$
Step 2:
Where $y=f(x)$
$\Rightarrow \lim\limits_{\large x\to a}g(f(x))=g(f(a))$
$x\to a\Rightarrow f(x)\to f(a)$
$\Rightarrow \lim\limits_{\large x\to a}gof(x)=gof(a)$
$gof$ is continuous at $x=a$
Hence it is a True statement.
answered Jul 5, 2013 by sreemathi.v
edited Jul 5, 2013 by sreemathi.v
 
Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...