If a function $f$ is continuous on $(-\infty ,\infty )$, which of the following statements is always true?

• A. $f$ is bounded on $(-\infty ,\infty )$
• B. $f$ has a limit at every real number
• C. $f$ is constant on $(-\infty ,\infty )$
• D. $f$ is differentiable at every real number

Answer: (B)

Verified step by step guidance

1. Step 1: Begin by understanding the concept of continuity. A function f is continuous on (−∞, ∞) if there are no breaks, jumps, or holes in its graph across the entire real number line.
2. Step 2: Analyze the first statement: 'f is bounded on (−∞, ∞)'. A continuous function is not necessarily bounded. For example, f(x) = x is continuous but unbounded. Therefore, this statement is not always true.
3. Step 3: Evaluate the second statement: 'f has a limit at every real number'. Continuity implies that the function approaches a specific value at every point in its domain, which means it has a limit at every real number. This statement is always true for a continuous function.
4. Step 4: Consider the third statement: 'f is constant on (−∞, ∞)'. Continuity does not imply that the function is constant. For example, f(x) = x is continuous but not constant. Thus, this statement is not always true.
5. Step 5: Examine the fourth statement: 'f is differentiable at every real number'. Continuity does not guarantee differentiability. For instance, f(x) = |x| is continuous but not differentiable at x = 0. Therefore, this statement is not always true.