Question

Hi, can someone help me with pset 1, question 1D-10? The question is:
Show that g(h)=[f(a+h)−f(a)]/h has a removable discontinuity at h=0 ⇐⇒ f'(a) exists.

Thank you!!

Answer 1

\( \exists f'(a) \Leftrightarrow \exists \lim_{h \to 0}g(h) \quad (\text{They are equivalent.}) \)
\( \exists \lim_{h \to 0}g(h) \Leftrightarrow \exists(\lim_{h \to 0^{+}}g(h) \wedge \lim_{h \to 0^{-}}g(h))\)
\( g(0) \neq \lim_{h \to 0}g(h) \Leftrightarrow g(x) \text{ at } 0 \text{ is a removable discontinuity. }\)
\( \therefore g(x) \text{ at } 0 \text{ is a removable discontinuity. } \Leftrightarrow \exists f'(a) \)

The essence of this proof is as follows: Since we know that \(g(0)\) is undefined, we can redefine the function at \(0\). We define that function to be equal to its limit as \(h\) approaches zero if and only if that limit exists. Since that limit is equivalent to \(f'(a)\) and \(f'(a)\) exists, the discontinuity can be removed.

Answer 2

Thank you!

Answer 3

You're welcome! :D