Examine the following functions for continuity. f( x)= x-5 - Vidyadip

Question

Examine the following functions for continuity.
$\text f(\text x)=\begin{vmatrix}\text x-5\end{vmatrix}$

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Answer

Here f(x) = | x - 5|
Function f is defined for all real numbers.
Let c be any real number.
$\therefore \text{f}(\text{c}) = \left| \text{c} - 5\right|$
Also $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\left|\text{x}- 5\right| = \left|\text{c}-5\right|$
$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = \text{f(c)}$
$\therefore$ f is continuous at x = c,
But c is any real number
$\therefore$ f is continuous at every real number.

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