Prove Continuity of Absolute Value Function

Prove that the function $f(x) = |x|$ is continuous on the real numbers using the epsilon-delta definition.

The challenge presented in this problem is to demonstrate the continuity of the absolute value function using the epsilon-delta definition of continuity. This task requires more than just knowing the abstract definition; it also requires the ability to apply this definition in a concrete, rigorous manner. The function $f(x) = |x|$ is defined for all real numbers and is a piecewise function, which introduces unique considerations when proving continuity using epsilon-delta techniques. Specifically, you need to show that for every positive epsilon, there exists a positive delta such that if the difference between x and a point c is less than delta, then the difference between $f(x)$ and $f(c)$ is less than epsilon. This requires analyzing the behavior of the function at $x = 0$, where the piecewise definition of absolute value changes, and ensuring continuity at that critical point as well as for all other real numbers.

Understanding continuity through the epsilon-delta definition is fundamental in real analysis as it forms the basis for more advanced concepts such as uniform continuity and differentiability. Mastering this type of problem builds a deeper understanding of the nature of continuous functions and prepares you for exploring more complex functions and theorems in the future. By being proficient with making epsilon-delta arguments, students gain insight into the meticulous nature of mathematical proofs and develop skills that are critical for success in higher-level mathematics.