Ask Questions, Get Answers

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

Examine the following functions for continuity. $f(x) = | \;x - 5\;|$

This is a multipart question answered separately on Clay6
Can you answer this question?

1 Answer

0 votes
  • If $f$ is a real function on a subset of the real numbers and $c$ a point in the domain of $f$, then $f$ is continous at $c$ if $\lim\limits_{x\to c} f(x) = f(c)$.
  • Every polynomial function $f(x)$ is continous.
Given $f(x) = | \;x - 5\;|$, $f(x) = x -5$, for $x \geq 5$ and $f(x) = 5-x$ for $x<5$.
$\textbf{Step 1}$:
The function $f$ is defined at all points of the real line.
Every polynomial function $f(x)$ is continous.
At $x >5$, $f(x) = x-5$ and at $x<5$, $f(x) = 5-x$, which are both polynomial functions, and $f(x)$ is continuous in both these cases.
$\textbf{Step 2}$:
At $x=5$, we need to evaluate the Right Hand Limit and Left Hand Limit and see if they are equal, in which case, $f(x)$ is continous.
$\Rightarrow$ RHL: For $x \to 5^+, \lim\limits_{x\to 5^+} f(x) = \lim\limits_{x\to 5^+} x-5 = 5-5 = 0$
$\Rightarrow$ LHL : For $x \to 5^-, \lim\limits_{x\to 5^-} f(x) = \lim\limits_{x\to 5^-} 5-x = 5-5 = 0$
Since RHL = LHL, $f$ is continuous in all cases.
answered Apr 4, 2013 by balaji.thirumalai

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App