The statement is True. The function $f(x) = |x|$ is indeed continuous at every real number. Absolute value functions do not have any breaks, jumps, or points of discontinuity. This continuity holds across all real numbers, including at $x = 0$, where there is a sharp corner but no discontinuity.

Would you like a detailed explanation on continuity or the concept of absolute value functions?

Here are some related questions to deepen your understanding:

1. What does it mean for a function to be continuous at a point?
2. How can you prove the continuity of a piecewise function?
3. Are all absolute value functions continuous?
4. What types of functions have points of discontinuity?
5. How does continuity differ from differentiability at sharp points, such as $x = 0$ for $f(x) = |x|$?

Tip: A function is continuous if you can draw it without lifting your pencil, meaning it has no gaps, jumps, or breaks in its domain.