Definition of Continuity at a Point:
A function f(x) is said to be continuous at a point x = a if the following three conditions are satisfied:

1. f(a) is defined (i.e., the function has a value at x = a).
2. The limit of f(x) as x approaches a exists, i.e.,

lim (x → a) f(x) exists.

3. The limit of f(x) as x approaches a is equal to the value of the function at that point, i.e.,

lim (x → a) f(x) = f(a).

In simple terms, for a function to be continuous at x = a, there should be no breaks, jumps, or holes in the graph of the function at that point.

Continuity of the Function f(x) = |x|:
The function f(x) = |x| is continuous for all real values of x. Here’s why:

1. f(x) = |x| is defined for all real values of x because the absolute value function is always defined.
2. The limit of f(x) as x approaches any value of x = a exists because both the left-hand limit (lim x → a⁻ f(x) and the right-hand limit (lim x → a⁺ f(x) exist and are equal.
3. The limit of f(x) as x approaches any value a is equal to the value of the function at that point. Specifically,

lim (x → a) |x| = |a|

Therefore, f(x) = |x| is continuous for all x ∈ R (the set of all real numbers). There is no point where the function is discontinuous.