We have a function $f:[a,b]\to \Bbb R$, where $[a,b]$ is an interval bounded by the real straight line. Now someone writes the definition of continuity at $x \in [a,b]$ but makes a mistake. He writes $\delta$ in stead of $\varepsilon$ and $\varepsilon$ instead of $\delta$. So the definition changes: for every $\delta > 0$ there exists $\varepsilon > 0$ such that if $x \in [a,b], |x - x_0| < \delta$ then $|f(x) - f(x_0)| < \varepsilon$.
Now we have to prove that this definition is satisfied if and only if the function is bounded by $[a,b]$.
How can I prove this?
Hint: Suppose you choose $\delta$ to be very large; say, larger than the length of the entire interval. What does the existence of $\epsilon$ tell you then?