1. **Problem Statement:** Determine if a function is continuous on a given interval. 2. **Key Concept:** A function $f(x)$ is continuous on an interval if it is continuous at every point in that interval. 3. **Continuity at a point $c$ means:** - $f(c)$ is defined. - The limit $\lim_{x \to c} f(x)$ exists. - $\lim_{x \to c} f(x) = f(c)$. 4. **To check continuity on an interval $[a,b]$:** - Check continuity at every point inside $(a,b)$. - Check right continuity at $a$ and left continuity at $b$. 5. **Example:** Suppose $f(x) = \frac{x^2 - 1}{x - 1}$ on $[0,2]$. 6. **Simplify $f(x)$:** $$f(x) = \frac{(x-1)(x+1)}{x-1}$$ 7. **Cancel common factor:** $$f(x) = \frac{\cancel{(x-1)}(x+1)}{\cancel{(x-1)}} = x + 1, \quad x \neq 1$$ 8. **Check continuity at $x=1$:** - $f(1)$ is not defined originally because denominator is zero. - But limit $\lim_{x \to 1} f(x) = 1 + 1 = 2$. 9. **Conclusion:** $f(x)$ is continuous everywhere on $[0,2]$ except at $x=1$ where it is not defined, so not continuous on the entire interval. 10. **To make $f$ continuous at $x=1$, define $f(1) = 2$. **Final answer:** $f(x)$ is continuous on $[0,2]$ except at $x=1$ unless $f(1)$ is defined as 2.