To prove that the function $f(x) = 5x - 3$ is continuous at $x = 0$, $x = -3$, and $x = 5$, we follow the definition of continuity at a point $c$:

A function $f(x)$ is continuous at $x = c$ if:

1. $f(c)$ is defined.
2. $\lim_{x \to c} f(x)$ exists.
3. $\lim_{x \to c} f(x) = f(c)$.

Step 1: Continuity at $x = 0$

1. Check if $f(0)$ is defined: $f(0) = 5(0) - 3 = -3.$ So, $f(0)$ is defined.

2. Evaluate ( \lim_{x \to 0} f(x): Since $f(x) = 5x - 3$ is a linear function, it is continuous everywhere in its domain, and the limit as $x \to 0$ is simply $f(0)$: $\lim_{x \to 0} f(x) = f(0) = -3.$

3. Verify if $\lim_{x \to 0} f(x) = f(0)$: $\lim_{x \to 0} f(x) = -3 = f(0).$ Hence, $f(x)$ is continuous at $x = 0$.

Step 2: Continuity at $x = -3$

1. Check if $f(-3)$ is defined: $f(-3) = 5(-3) - 3 = -15 - 3 = -18.$ So, $f(-3)$ is defined.

2. Evaluate ( \lim_{x \to -3} f(x): Again, since $f(x) = 5x - 3$ is linear, the limit as $x \to -3$ is simply $f(-3)$: $\lim_{x \to -3} f(x) = f(-3) = -18.$

3. Verify if $\lim_{x \to -3} f(x) = f(-3)$: $\lim_{x \to -3} f(x) = -18 = f(-3).$ Hence, $f(x)$ is continuous at $x = -3$.

Step 3: Continuity at $x = 5$

1. Check if $f(5)$ is defined: $f(5) = 5(5) - 3 = 25 - 3 = 22.$ So, $f(5)$ is defined.

2. Evaluate ( \lim_{x \to 5} f(x): Since $f(x) = 5x - 3$ is linear, the limit as $x \to 5$ is simply $f(5)$: $\lim_{x \to 5} f(x) = f(5) = 22.$

3. Verify if $\lim_{x \to 5} f(x) = f(5)$: $\lim_{x \to 5} f(x) = 22 = f(5).$ Hence, $f(x)$ is continuous at $x = 5$.

Conclusion:

The function $f(x) = 5x - 3$ is continuous at $x = 0$, $x = -3$, and $x = 5$.

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Do you want more details or have further questions?

Related Questions:

1. What is the general proof that linear functions are continuous everywhere?
2. How do we handle continuity proofs for piecewise functions?
3. Can a function be discontinuous even if it’s defined at a point?
4. What’s the difference between removable and essential discontinuities?
5. How can we prove continuity using the $\epsilon$-$\delta$ definition?

Tip:

For polynomial and linear functions, continuity can often be assumed directly since they are continuous on their entire domain.