Questions: Prove that f(x) is continuous at c if and only if the limit as h approaches 0 of f(c+h)=f(c). First, assume f(x) is continuous at c. From the definition of continuity, what is known about the function? - The limit as x approaches c of f(x) exists - The limit as x approaches c of f(x)=f(c) - f(c) exists - All of the above. If the limit as x approaches c of f(x)=f(c), how can this be used to show the limit as h approaches 0 of f(c+h)=f(c)? - Define h equal to x-c. - This cannot be shown. - Define h equal to x+c. - Since h approaches zero, the statements are not equivalent. Now, assume the limit as h approaches 0 of f(c+h)=f(c). Can h be defined so that f(c) and the limit as x approaches c of f(x) exist? - Yes - No

Prove that f(x) is continuous at c if and only if the limit as h approaches 0 of f(c+h)=f(c).

First, assume f(x) is continuous at c. From the definition of continuity, what is known about the function?
- The limit as x approaches c of f(x) exists
- The limit as x approaches c of f(x)=f(c)
- f(c) exists
- All of the above.
If the limit as x approaches c of f(x)=f(c), how can this be used to show the limit as h approaches 0 of f(c+h)=f(c)?
- Define h equal to x-c.
- This cannot be shown.
- Define h equal to x+c.
- Since h approaches zero, the statements are not equivalent.
Now, assume the limit as h approaches 0 of f(c+h)=f(c). Can h be defined so that f(c) and the limit as x approaches c of f(x) exist?
- Yes
- No

Transcript text: Prove that $f(x)$ is continuous at $c$ if and only if $\lim _{h \rightarrow 0} f(c+h)=f(c)$. First, assume $f(x)$ is continuous at $c$. From the definition of continuity, what is known about the function? $\lim _{x \rightarrow c} f(x)$ exists $\lim _{x \rightarrow c} f(x)=f(c)$ $f(c)$ exists All of the above. If $\lim _{x \rightarrow c} f(x)=f(c)$, how can this be used to show $\lim _{h \rightarrow 0} f(c+h)=f(c)$ ? Define $h$ equal to $x-c$. This cannot be shown. Define $h$ equal to $x+c$. Since $h$ approaches zero, the statements are not equivalent. Now, assume $\lim _{h \rightarrow 0} f(c+h)=f(c)$. Can $h$ be defined so that $f(c)$ and $\lim _{x \rightarrow c} f(x)$ exist? Yes No

Solution

Solution Steps

Step 1: Understanding the definition of continuity

From the definition of continuity, if $ f(x) $ is continuous at $ c $, then:

$ \lim_{x \rightarrow c} f(x) $ exists.
$ \lim_{x \rightarrow c} f(x) = f(c) $.
$ f(c) $ exists.

Thus, all of the above statements are true.

Step 2: Relating $ \lim_{x \rightarrow c} f(x) = f(c) $ to $ \lim_{h \rightarrow 0} f(c+h) = f(c) $

If $ \lim_{x \rightarrow c} f(x) = f(c) $, we can define $ h = x - c $. As $ x \rightarrow c $, $ h \rightarrow 0 $. Substituting $ x = c + h $ into the limit gives: \[ \lim_{h \rightarrow 0} f(c + h) = f(c). \]

Step 3: Proving the converse

Assume $ \lim_{h \rightarrow 0} f(c + h) = f(c) $. Let $ x = c + h $. As $ h \rightarrow 0 $, $ x \rightarrow c $. Substituting $ h = x - c $ into the limit gives: \[ \lim_{x \rightarrow c} f(x) = f(c). \] This shows that $ f(c) $ exists and $ \lim_{x \rightarrow c} f(x) = f(c) $, which are the conditions for continuity at $ c $.

Final Answer

The correct answer is D.

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