The problem asks you to prove that a function $f$ is continuous at $c$ if and only if:

$\lim_{t \to 0} f(c + t) = f(c)$

Proof Outline:

Definition of Continuity at $c$:

A function $f$ is said to be continuous at a point $c$ if: $\lim_{x \to c} f(x) = f(c)$ In this case, $f(x)$ must approach $f(c)$ as $x$ gets arbitrarily close to $c$.

Given Statement:

We are given that continuity at $c$ holds if and only if: $\lim_{t \to 0} f(c + t) = f(c)$

This expression essentially shifts $x$ near $c$ by writing $x = c + t$, and investigates what happens as $t$ tends to 0. This is equivalent to checking the limit as $x \to c$.

Proof:

1. If $f$ is continuous at $c$, then: By definition, continuity at $c$ implies: $\lim_{x \to c} f(x) = f(c)$ Now, let $x = c + t$, which means $x \to c$ is the same as $t \to 0$. Thus, the limit becomes: $\lim_{t \to 0} f(c + t) = f(c)$ This shows that if $f$ is continuous at $c$, then the given condition holds.

2. If $\lim_{t \to 0} f(c + t) = f(c)$, then $f$ is continuous at $c$: Suppose the limit $\lim_{t \to 0} f(c + t) = f(c)$ holds. Now, as $t \to 0$, $c + t \to c$, meaning that $f(x)$ approaches $f(c)$ as $x$ approaches $c$. This satisfies the definition of continuity at $c$, and therefore, $f$ is continuous at $c$.

Conclusion:

The statement is proven. A function $f$ is continuous at $c$ if and only if: $\lim_{t \to 0} f(c + t) = f(c)$

Would you like more details or have any questions about this proof?

Here are some related questions for further exploration:

1. What is the definition of a limit in mathematical terms?
2. Can you explain the difference between continuity and differentiability?
3. How does continuity relate to intermediate value theorem?
4. What are the different types of discontinuities?
5. How would you prove the continuity of a piecewise function?

Tip: Understanding the concept of limits is crucial in grasping the definition of continuity, especially when analyzing different types of limits (one-sided, two-sided).