Jelly, 8 bytes

1DI*`ƑƊ#

A full program accepting an integer, n, from STDIN which prints a list of the first n positive jumping numbers.

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How?

Acceptable incremental differences between digits are 1 and -1 while others from [-9,-2]+[2,9] are not. This lines up with integers which are invariant when raised to themselves. i.e. $x^x=x$ since:

$$0^0=1$$
$$1^1=1$$
$$2^2=4$$
$$cdots$$
$$-1^{-1}=-1$$
$$-2^{-2}=-frac{1}{4}$$
$$cdots$$

1DI*`ƑƊ# - Main Link: no arguments (accepts a line of input from STDIN)

       # - count up keeping the first (input) n matches...

1        - ...start with n equal to: 1

      Ɗ  - ...match function: last three links as a monad:  e.g. 245       777      7656

 D       -   convert to a list of decimal digits                 [2,4,5]   [7,7,7]  [7,6,5,6]

  I      -   incremental differences                             [2,1]     [0,0]    [-1,-1,1]

     Ƒ   -   invariant under?:

    `    -     using left argument as both inputs of:

   *     -       exponentiation (vectorises)                     [4,1]     [1,1]    [-1,-1,1]

         -                                            --so we:   discard   discard  keep

         - implicitly print the list of collected values of n

05AB1E (legacy), 5 bytes

The input is 1-indexed.

Code:

µN¥ÄP

Uses the 05AB1E encoding. Try it online!

Explanation

µ          # Get the nth number, starting from 0, such that...

   Ä       #   The absolute values

 N¥        #   Of the delta's of N

    P      #   Are all 1 (product function, basically acts as a reduce by AND)

C (gcc), 90 bytes

f(n,K,b,k){for(K=0;n;b&&printf("%d,",K,n--))for(b=k=++K;k/10;)b*=abs(k%10-(k/=10)%10)==1;}

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Japt, 14 bytes

Outputs the first nth term, 1-indexed.

_ì äa dÉ ªU´}f

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(I know, I know, I'm supposed to be taking a break but I'm in golf withdrawal!)

Python 2, 88 87 bytes

f=lambda n,i=2:n and f(n-g(i),i+1)or~-i

g=lambda i:i<10or abs(i/10%10-i%10)==1==g(i/10)

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Returns the 0-indexed jumping number (i.e., f(0) => 1, etc).

JavaScript (ES7), 60 bytes

Returns the $n$^th term of the sequence (1-indexed).

f=(n,k)=>[...k+''].some(p=x=>(p-(p=x))**2-1)||n--?f(n,-~k):k

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Jelly, 9 bytes

-1 by Jonathan Allan

1DạƝ=1ẠƲ#

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1-indexed.

Haskell, 79 bytes

• Thanks to Joseph Sible for enforcing the challenge rules.

f=(filter g[1..]!!)

g m|d<-read.pure<$>show m=all((==1).abs).zipWith(-)d$tail d

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Haskell, 57 bytes

(l!!)

l=[1..9]++[x*10+t|x<-l,t<-[0..9],(mod x 10-t)^2==1]

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Swift, 228 bytes

func j(n:Int){

var r:[Int]=

for x in 0...n{

if x<=10{r.append(x)}else{

let t=String(x).compactMap{Int(String($0))}

var b=true

for i in 1...t.count-1{if abs(t[i-1]-t[i]) != 1{b=false}}

if b{r.append(x)}

}

}

print(r)

}

j(n:1000)

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Python 3, 122 121 bytes

g=lambda s:len(s)==1or 1==abs(ord(s[0])-ord(s[1]))and g(s[1:])

def f(n,i=1):

	while n:

		if g(str(i)):n-=1;yield i

		i+=1

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-1 byte by changing f from printing, to a generator function.

g is a recursive helper function which determines if a string s is a "jumping string" (this works since the character codes for 0 to 9 are in order and contiguous).

f is a generator function that takes in n and yields the first n jumping numbers.

R, 85 bytes

i=scan();j=0;while(i)if((j=j+1)<10|all(abs(diff(j%/%10^(0:log10(j))%%10))==1))i=i-1;j

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Suspect this can be golfed more. Reads the number using scan() and outputs the appropriate jumping number.

Perl 5, 56 bytes

map{1while++$n=~s|.(?=(.))|abs$&-$1!=1|ger>9}1..<>;say$n

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1-indexed, outputs the n^th jumping number

Wolfram Language (Mathematica), 85 bytes

If[#<10,#,t=n=1;While[t<=#,{-1,1}~SubsetQ~Differences@IntegerDigits@n++~If~t++];n-1]&

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returns the n-th number

Python 2, 79 bytes

f=lambda n,i=2:n and f(n-g(i),i+1)or~-i

g=lambda i:i<10or i%100%11%9==g(i/10)>0

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Derived from Chas Brown's answer. The helper function g(i) returns whether i is a jumping number. Iff the last two digits of a number n have absolute difference 1, then n%100%11 will be either 1 or 10, so n%100%11%9 will be 1.