Here are some MIT-style questions on continuity, suitable for enhancing understanding of the topic. These questions explore fundamental principles of continuity, real analysis, and calculus at a higher level.

Questions on Continuity

1. Limits and Continuity
   Let $f(x)$ be defined as follows:

   \frac{\sin(x)}{x} & x \neq 0 \\ 1 & x = 0 \end{cases}$$ Is $$f(x)$$ continuous at $$x = 0$$? Justify your answer by evaluating the limit as $$x \to 0$$ and comparing it to $$f(0)$$.

2. Piecewise Continuous Functions
   Define a function $g(x)$ as:

   x^2 & x \leq 1 \\ 2 - x & x > 1 \end{cases}$$ Determine if $$g(x)$$ is continuous at $$x = 1$$. If it is not continuous, determine the type of discontinuity present and explain how you can modify $$g(x)$$ to make it continuous at $$x = 1$$.

3. Continuity and Intermediate Value Theorem
   Let $h(x) = x^3 - 3x + 1$. Prove that there exists a point $c \in (0, 1)$ such that $h(c) = 0$ using the Intermediate Value Theorem (IVT). Ensure that your answer includes a verification of continuity on the interval $[0,1]$.

4. Continuity of Composite Functions
   Suppose $f(x)$ is a continuous function on $\mathbb{R}$ and $g(x)$ is defined as:

   f\left( \frac{1}{x} \right) & x \neq 0 \\ 0 & x = 0 \end{cases}$$ Under what conditions on $$f(x)$$ is $$g(x)$$ continuous at $$x = 0$$? Provide a proof or counterexample.

5. Uniform Continuity
   Consider the function $f(x) = \frac{1}{x}$ on the interval $(0, 1]$. Is $f(x)$ uniformly continuous on this interval? Prove your answer by either using the definition of uniform continuity or providing a counterexample.

Tip

When studying continuity, it's helpful to understand the definitions and properties of pointwise versus uniform continuity, as this distinction often appears in real analysis and advanced calculus contexts.

Would you like details on any specific question or have any other questions? Here are five questions that expand on continuity:

1. How does differentiability imply continuity, and are there exceptions?
2. Can a function be continuous at a point without being differentiable there? Provide examples.
3. What is the significance of removable and essential discontinuities in real analysis?
4. How does the epsilon-delta definition apply to proving continuity at a point?
5. In what ways do the concepts of left-hand and right-hand limits contribute to understanding continuity?