## JingleBells 3 years ago Problem Set 1: 1D-10 Show that

1. JingleBells

$g(h)=\left[ f(a+h)-f(a) \right]/h$has a removable discontinuity at h=0 $f \prime(a) exist$

2. JingleBells

I gave it a try, is anyone convinced? $g(0^{-})=\lim_{h \rightarrow 0}\left[ f(a+h)-f(a) \right]/h=f \prime (a)$ $g(0^{+})=\lim_{h \rightarrow 0}\left[ f(a+h)-f(a) \right]/h=f \prime (a)$ Since the left-hand limit=right-hand limit at h=0, i.e. $f \prime(a)=f \prime(a)$ Therefore f'(a) exists and g(h) has a removable discontinuity at h=0 Sounds about right? seems too simple an answer...

3. JXP

Yes, it is correct. The first problem set goes over some basics of high school calculus since some of the students have not taken calculus yet. So keep on going if you're fairly confident with your answer!

4. Stacey

What you have is correct. It is removable because by defining g(0) = f'(a) the function becomes continuous at h=0.

5. JingleBells

Thanks a million! I'm using these videos to prepare for A-levels so I'm bound to be out of my depth most of the time.