Let f: ℝ →ℝ be continuous. (a) If f(0)>0, show that f(x)>0...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. (a) If $f(0)>0$, show that $f(x)>0$ for all $x$ in some open interval $(-a, a)$. (b) If $f(x) \geq 0$ for every rational $x$, show that $f(x) \geq 0$ for all real $x$. Will this result hold with " $\geq 0$ " replaced by " $>0$ "? Explain.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous.
(a) If $f(0)>0$, show that $f(x)>0$ for all $x$ in some open interval $(-a, a)$.
(b) If $f(x) \geq 0$ for every rational $x$, show that $f(x) \geq 0$ for all real $x$. Will this result hold with " $\geq 0$ " replaced by " $>0$ "? Explain.

Key Concepts

Continuity

Continuity is the property of a function which guarantees that small changes in the input lead to small changes in the output. This means that, at any point in its domain, the function does not have sudden jumps or breaks. The concept is formalized via the epsilon-delta definition, ensuring that if a function value is positive at a particular point, nearby values of the function remain close to that positive number.

Open Intervals and Neighborhoods

An open interval or neighborhood around a point is a set of points within a certain distance from that point where specific properties hold. In the context of continuous functions, if a function has a positive value at a point, continuity ensures that there exists an open interval (or neighborhood) around that point where the function’s values stay positive, because deviations of the input are controlled to make the output deviate only slightly from the given positive value.

Density of Rational Numbers

The rational numbers are dense in the real numbers, meaning that within any interval of real numbers there are always rational numbers. This concept is critical in problems where a property is known to hold for all rational numbers; under continuity, one can often extend such properties to all real numbers. However, the density property may not help when dealing with strict inequalities due to the potential of the function taking a boundary value (like zero) at irrational points.

Counterexample Reasoning in Continuous Functions

Counterexample reasoning is used to show the limitations of certain conclusions under modified assumptions. For instance, while the non-negativity of a continuous function on the rationals can extend to all real numbers by density and continuity, this extension may fail for strict positivity. A counterexample can be constructed where a function is strictly positive on rationals but attains zero on irrationals, illustrating that the property does not hold if the inequality is strengthened.

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