If $f(x) = |x| + |x - 5|$, then which of the following statements are TRUE?

(A) f is a continuous function every where

(B) f is a continuous function except $x = 5$ and $x = 0$

(C) f is a continuous function except $x = 0$ but not differentiable at $x = 5$

(D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$

Choose the correct answer from the options given below:

The function $f(x) = |x| + |x - 5|$ changes behavior at $x = 0$ and $x = 5$.

For $x < 0$:

$|x| = -x$

$|x - 5| = -x + 5$

$f(x) = -2x + 5$

For $0 \leq x < 5$:

$|x| = x$

$|x - 5| = -x + 5$

$f(x) = 5$

For $x \geq 5$:

$|x| = x$

$|x - 5| = x - 5$

$f(x) = 2x - 5$

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At $x = 0$:

$f(0^-) = -2(0) + 5 = 5$

$f(0^+) = 5$

$f(0) = 5$

The function is continuous at $x = 0$.

At $x = 5$:

$f(5^-) = 5$

$f(5^+) = 2(5) - 5 = 5$

$f(5) = 5$

The function is continuous at $x = 5$.

Therefore, $f$ is continuous everywhere.

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At $x = 0$:

Left derivative: $\frac{d}{dx}(-2x + 5) = -2$

Right derivative: $\frac{d}{dx}(5) = 0$

The derivatives are not equal, so $f$ is not differentiable at $x = 0$.

At $x = 5$:

Left derivative: $\frac{d}{dx}(5) = 0$

Right derivative: $\frac{d}{dx}(2x - 5) = 2$

The derivatives are not equal, so $f$ is not differentiable at $x = 5$.

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(A) $f$ is a continuous function everywhere → TRUE

(B) $f$ is a continuous function except $x = 5$ and $x = 0$ → FALSE

(C) $f$ is a continuous function except $x = 0$ but not differentiable at $x = 5$ → FALSE

(D) $f$ is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$ → TRUE

Statements (A) and (D) are TRUE.