×

# Suppose f(x) is even function. if f(x) is continous at a, show f(x) continuous at -a ?

Feb 24, 2018

See below

#### Explanation:

The definition of an even function is $f \left(- x\right) = f \left(x\right)$

Therefore, $f \left(- a\right) = f \left(a\right)$. Since $f \left(a\right)$ is continuous and $f \left(- a\right) = f \left(a\right)$, then $f \left(- a\right)$ is also continuous.

Mar 9, 2018

Check below for detailed solution

#### Explanation:

• $f$ even means: for each $x$$\in$$\mathbb{R}$ , $- x$$\in$$\mathbb{R}$

$f \left(- x\right) = f \left(x\right)$

• $f$ continuous at ${x}_{0} = a$ $\iff$ ${\lim}_{x \to a} f \left(x\right) = f \left(a\right)$

${\lim}_{x \to - a} f \left(x\right)$

Set $y = - x$
$x \to - a$
$y \to a$

$=$ ${\lim}_{y \to a} f \left(- y\right) = {\lim}_{y \to a} f \left(y\right) = {\lim}_{x \to a} f \left(x\right) = f \left(a\right)$